Given:

a1x + b1y + c1 = 0..............................(1)

a2x + b2y + c2 = 0..............................(2)

Slope of eqn (1), m1 = - $$\frac {x-coefficient}{y-coefficient}$$ = -$$\frac {a_1}{b_1}$$

Slope of eqn (2), m2 = - $$\frac {x-coefficient}{y-coefficient}$$ = -$$\frac {a_2}{b_2}$$

when the lines are parallel, then:

m1 = m2

or,-$$\frac {a_1}{b_1}$$ =-$$\frac {a_2}{b_2}$$

∴ a1b2 = a2b1 Ans

when the lines are perpendicular, then:

m1× m2= - 1

or,-$$\frac {a_1}{b_1}$$×-$$\frac {a_2}{b_2}$$ = - 1

∴ a1a2 = - b1b2Ans

Here,

5x + 4y - 10 = 0............................(1)

15x + 12y - 7 = 0.........................(2)

Slope of eqn (1), m1 = - $$\frac {x-coefficient}{y-coefficient}$$ = -$$\frac 54$$

Slope of eqn (2), m2 = - $$\frac {x-coefficient}{y-coefficient}$$ = -$$\frac {15}{12}$$ = -$$\frac 54$$

m1 = m2 = - $$\frac 54$$

∴ The given two lines are parallel to each other. Proved

Here,

x + 3y = 2..................................(1)

6x - 2y = 9................................(2)

Slope of eqn (1), m1 = - $$\frac {x-coefficient}{y-coefficient}$$ = - $$\frac 13$$

Slope of eqn (2), m2 = - $$\frac {x-coefficient}{y-coefficient}$$ = - $$\frac 6{-2}$$ =3

We have,

m1× m2 =- $$\frac 13$$× 3 = - 1

∴m1× m2 = - 1

Hence, the given two lines are perpendicular each other. Proved

Here,

3x - 2y - 5 = 0..............................(1)

2x + py - 3 = 0.............................(2)

Slope of eqn (1), m1 = - $$\frac {x-coefficient}{y-coefficient}$$ = - $$\frac 3{-2}$$ = $$\frac 32$$

Slope of eqn (2), m2 = - $$\frac {x-coefficient}{y-coefficient}$$ = - $$\frac 2{p}$$

When two lines are parallel lines, then:

m1 = m2

or,$$\frac 32$$ = - $$\frac 2{p}$$

or, p = $$\frac {-2 × 2}3$$

∴ p = -$$\frac 43$$ Ans

Here,

4x + ky - 4 = 0..............................(1)

2x - 6y = 5.............................(2)

Slope of eqn (1), m1 = - $$\frac {x-coefficient}{y-coefficient}$$ = - $$\frac 4k$$

Slope of eqn (2), m2 = - $$\frac {x-coefficient}{y-coefficient}$$ = - $$\frac 2{-6}$$ = $$\frac 13$$

when two line are perpendicular to each other,

m1× m2 = - 1

or, - $$\frac 4k$$ × $$\frac 13$$ = - 1

or, - $$\frac 43$$× - 1 = k

∴ k = $$\frac 43$$ Ans

The formulae of angle between y = m1x + c1 andy = m2x + c2is:

tan$$\theta$$ = ± $$\frac {m_1 - m_2}{1 + m_1m_2}$$

when m1× m2 = -1, the two lines are perpendicular to each other.

when m1 = m2, the two lines are parallel to each other.

Here,

Slope of points (3, -4) and (-2, a)

m1 = $$\frac {y_2 - y_1}{x_2 - x_1}$$ = $$\frac {a + 4}{-2 - 3}$$ = $$\frac {-(a + 4)}5$$

Given eqn is y + 2x + 3 = 0

Slope of above eqn (m2) = - $$\frac {x-coefficient}{y-coefficient}$$ = -$$\frac 21$$ = - 2

when lines are parallel then,

m1 = m2

or,$$\frac {-(a + 4)}5$$ = - 2

or, a + 4 = 10

or, a = 10 - 4

∴ a = 6 Ans

Here,

Slope of the points (3, -4) and (-2, 6) is:

m1 = $$\frac {y_2 - y_1}{x_2 - x_1}$$ = $$\frac {6 + 4}{-2 - 3}$$ = $$\frac {10}{-5}$$ = -2

Slope of the eqn y + 2x + 3 = 0 is:

m2 = - $$\frac {x-coefficient}{y-coefficient}$$ = - $$\frac 21$$ = -2

From above,

m1= m2 = - 2

Hence, the lines are parallel. Proved

Here,

Given eqn is kx - 3y + 6 = 0

Slope of above eqn is:

m1 = - $$\frac {x-coefficient}{y-coefficient}$$ = - $$\frac k{-3}$$ = $$\frac k3$$

Slope ofthe point (4, 3) and (5, -3) is:

m2 = $$\frac {y_2 - y_1}{x_2 - x_1}$$ = $$\frac {-3 - 3}{5 - 4}$$ = - $$\frac 61$$ = -6

If lines are perpendicular then:

m1× m2 = -1

or, $$\frac k3$$× -6 = -1

or, k = $$\frac {-1}{-2}$$

∴ k = $$\frac 12$$ Ans

Here,

Given equations of the lines are:

y = m1x + c1...............................(1)

y = m2x + c2...............................(2)

If $$\theta$$ be the angle between two lines (1) and (2);

The formula of angle between the given lines is:

tan$$\theta$$ =± $$\frac {m_1 - m_2}{1 + m_1m_2}$$

∴ $$\theta$$ = tan-1(± $$\frac {m_1 - m_2}{1 + m_1m_2}$$)

If two lines are perpendicular ($$\theta$$ = 90°)

tan 90° =± $$\frac {m_1 - m_2}{1 + m_1m_2}$$

or,∞=± $$\frac {m_1 - m_2}{1 + m_1m_2}$$

or, $$\frac 10$$ =± $$\frac {m_1 - m_2}{1 + m_1m_2}$$

or, 1 + m1m2 = 0

∴ m1m2 = -1 Ans

Here,

The given equations are:

2x + ay + 3 = 0....................(1)

3x - 2y = 5.............................(2)

Slope of equation (1), m1 = -$$\frac {x-coefficient}{y-coefficient}$$ = - $$\frac 2a$$

Slope of equation (2), m2 = -$$\frac {x-coefficient}{y-coefficient}$$ = - $$\frac 3{-2}$$ = $$\frac 32$$

If equation (1) and equation (2) are perpendicular to each other:

m1× m2 = -1

or,- $$\frac 2a$$×$$\frac 32$$ = -1

or, -6 = - 2a

or, a = $$\frac 62$$

∴ a = 3 Ans

Here,

Given equations of the lines are:

2x + 4y - 7 = 0...........................(1)

6x + 12y + 4 = 0.......................(2)

Slope of equation (1) is: m1 = - $$\frac {x-coefficient}{y-coefficient}$$ = -$$\frac 24$$ = -$$\frac 12$$

Slope of equation (2) is: m2 = - $$\frac {x-coefficient}{y-coefficient}$$ = -$$\frac 6{12}$$ = -$$\frac 12$$

∴ m1 = m2 = $$\frac {-1}2$$

Since, the slope of these equations are equal, the lines are parallel to each other. Proved

Given lines are:

3x + 5y = 7 i.e. 3x + 5y - 7 = 0........................(1)

3y = 2x + 4 i.e. 2x - 3y + 4 = 0........................(2)

Slope of eqn (1), m1 = - $$\frac {x-coefficient}{y-coefficient}$$ = $$\frac {-2}{-3}$$ = $$\frac 23$$

Slope of eqn (2), m2 = - $$\frac {x-coefficient}{y-coefficient}$$ = $$\frac {-3}{5}$$

Let $$\theta$$ be the angle between the equation (1) and (2):

tan$$\theta$$ =± $$\frac {m_1 - m_2}{1 + m_1m_2}$$

or, tan$$\theta$$ =± ($$\frac {\frac 23 + \frac 35}{1 - \frac 23 × \frac 35}$$)

or, tan$$\theta$$ =± ($$\frac {\frac {10 + 9}{15}}{\frac {5 - 2}5}$$)

or, tan$$\theta$$ =± ($$\frac {19}{15}$$ × $$\frac 53$$)

or, tan$$\theta$$ =± $$\frac {19}9$$

For acute angle,

tan$$\theta$$ = $$\frac {19}9$$= 2.11

∴ $$\theta$$ = 65°

∴ The acute angle between two lines is 65°. Ans

Here,

Given equation are:

3y - x - 6 = 0..............................(1)

y = 2x + 5 i.e. -2x + y = 5...................(2)

Slope of eqn (1), m1= -$$\frac {x-coefficient}{y-coefficient}$$ = -$$\frac {(-1)}3$$ = $$\frac 13$$

Slope of eqn (2), m2= -$$\frac {x-coefficient}{y-coefficient}$$ = -$$\frac {(-2)}1$$ = 2

If $$\theta$$ be the angle between the eqn (1) and (2),

tan$$\theta$$ =± $$\frac {m_1 - m_2}{1 + m_1m_2}$$

or, tan$$\theta$$ =± $$\frac {\frac 13 - 2}{1 + \frac 13 × 2}$$

or, tan$$\theta$$ =± $$\frac {\frac {1 - 6}3}{\frac {3 + 2}3}$$

or, tan$$\theta$$ =± $$\frac {-5}3$$× $$\frac 35$$

∴ tan$$\theta$$ =± (-1)

Taking -ve sign,

tan$$\theta$$ = +1

tan$$\theta$$ = 45°

∴$$\theta$$ = 45° Ans

Here,

x - 3y = 4......................(1)

2x - y = 3......................(2)

Slope of eqn (1), m1 = -$$\frac {x-coefficient}{y-coefficient}$$ = $$\frac {-1}{-3}$$ = $$\frac 13$$

Slope of eqn (2), m2 = -$$\frac {x-coefficient}{y-coefficient}$$ = $$\frac {-2}{-1}$$ = 2

If $$\theta$$ be the angle between the eqn (1) and (2),

tan$$\theta$$ =± $$\frac {m_1 - m_2}{1 + m_1m_2}$$

or, tan$$\theta$$ =± $$\frac {\frac 13 - 2}{1 + \frac 13 × 2}$$

or, tan$$\theta$$ =± $$\frac {\frac {1 - 6}3}{\frac {3 + 2}3}$$

or, tan$$\theta$$ =± $$\frac {-5}3$$× $$\frac 35$$

∴ tan$$\theta$$ =± (-1)

Taking -ve sign,

tan$$\theta$$ = +1

tan$$\theta$$ = 45°

∴$$\theta$$ = 45° Ans

Here,

Given equation are:

y - 3x - 2 = 0

or, -3x + y - 2 = 0..............................(1)

y = 2x + 5

or, - 2x + y = 5...................................(2)

Slope of eqn (1), m1 = -$$\frac {x-coefficient}{y-coefficient}$$ = $$\frac {-3}{-1}$$ = 3

Slope of eqn (2), m2 = -$$\frac {x-coefficient}{y-coefficient}$$ = -$$\frac {-2}{1}$$ =2

If $$\theta$$ be the angle between two lines,

tan$$\theta$$ =± $$\frac {m_1 - m_2}{1 + m_1m_2}$$

or, tan$$\theta$$ =± $$\frac {3 - 2}{1 + 3 × 2}$$

or, tan$$\theta$$ =± $$\frac 1{1 + 6}$$

∴ tan$$\theta$$ =± $$\frac 17$$

Taking +ve sign,

$$\theta$$ = tan-1($$\frac 17$$)

∴ $$\theta$$ = 8.13° Ans

Here,

Given lines are:

x = 3y + 8

i.e. x - 3y - 8 = 0..................................(1)

2x + 11 = 7y

i.e. 2x - 7y + 11 = 0............................(2)

Slope of eqn (1),m1= - $$\frac {x-coefficient}{y-coefficient}$$ = - $$\frac 1{-3}$$ = $$\frac 13$$

Slope of eqn (2),m2= - $$\frac {x-coefficient}{y-coefficient}$$ = - $$\frac 2{-7}$$ = $$\frac 27$$

Let $$\theta$$ be the angle between the equation (1) and (2),

tan$$\theta$$ =± $$\frac {m_1 - m_2}{1 + m_1m_2}$$

or, tan$$\theta$$ = ± $$\frac {\frac 13 - \frac 27}{1 + \frac 13 × \frac 27}$$

or, tan$$\theta$$ =± $$\frac {\frac {7 - 6}{21}}{\frac {21 + 2}{21}}$$

or, tan$$\theta$$ =± $$\frac 1{21}$$× $$\frac {21}{23}$$

∴ tan$$\theta$$ =± $$\frac 1{23}$$

For obtuse angle,

tan$$\theta$$ = -$$\frac 1{23}$$

or, tan$$\theta$$ = tan (180° -2°)

∴ $$\theta$$ = 178°

∴ The obtuse angle between two lines is 178°. Ans

Given equation is:

5x - 7y + 10 = 0............................(1)

Equation (1) changes in perpendicular form

7x + 5y + k = 0.............................(2)

The point (2, -1) passes through the equation (2)

7× 2 + 5× -1 + k = 0

or, 14 - 5 + k = 0

or, 9 + k = 0

∴ k = -9

Putting the value of k in equation (2)

7x + 5y - 9 = 0

∴ Required equation is:7x + 5y - 9 = 0 Ans

Given line is:

5x - 4y + 19 = 0...........................(1)

Equation (1) changes into the perpendicular form:

4x + 5y + k = 0............................(2)

The point (2, -3) passes through the equation (2)

4× 2 + 5× -3 + k = 0

or, 8 - 15 + k = 0

or, -7 + k = 0

∴ k = 7

Putting the value of k in equation (2)

4x + 5y + 7 = 0

∴ The required equation is 4x + 5y + 7 = 0. Ans

The slope of the line joining (3, - 7) and (-5, 3) is:

m1 = $$\frac {y_2 - y_1}{x_2 - x_1}$$ = $$\frac {3 + 7}{-5 - 3}$$ = $$\frac {10}{-8}$$ = -$$\frac 54$$

The mid-pointof the line joining (3, - 7) and (-5, 3) is:

P(x, y) = ($$\frac {3 - 5}2$$, $$\frac {-7 + 3}2$$) = ($$\frac {-2}2$$, $$\frac {-4}2$$) = (-1, -2)

The equation of the line passing through (-1, -2) is:

y + 2 = m (x + 1)............................(1)

Equation (1) is the perpendicular bisector of the line joining given two points,

m1× m2 = -1

or,-$$\frac 54$$× m2 = -1

∴ m2 = $$\frac 45$$

Putting the value of m2 in equation (1)

y + 2 = $$\frac 45$$(x + 1)

or, 5y + 10 = 4x + 4

or, 4x - 5y - 10 + 4 = 0

∴ 4x - 5y - 6 = 0 Ans

Here,

Given line is:

3x - 4y + 9 = 0..........................(1)

The line perpendicular to the equation (1) is:

4x + 3y + k = 0.........................(2)

The equation (2) passes through the point (-6, 4)

4× -6 + 3× 4 + k = 0

or, -24 + 12 + k = 0

or, -12 + k = 0

∴ k = 12

Putting the value of k in equation (2)

4x + 3y + 12 = 0 Ans

Given:

The co-ordinatesof A and B are: (3, -1) and (7, 1) respectively.

Equation of A(3, -1) and B(7, 1):

y + 1 = $$\frac {1 + 1}{7 - 3}$$ (x - 3) [$$\because$$ y - y1 = $$\frac {y_2 - y_1}{x_2 - x_1}$$ (x - x1)]

or, 4y + 4 = 2x - 6

or, 2x - 4y - 6 - 4 = 0

or, 2x - 4y - 10 = 0

or, 2(x - 2y - 5) = 0

∴ x - 2y - 5 = 0...................................(1)

Mid-point of A(3, -1) and B(7, 1)

= ($$\frac {3 + 7}2$$, $$\frac {-1 + 1}2$$) [$$\because$$ (x, y) = ($$\frac {x_1 + x_2}2$$, $$\frac {y_1 + y_2}2$$)]

= ($$\frac {10}2$$, $$\frac 02$$)

= (5, 0)

The perpendicular form of the equation x - 2y - 5 = 0 is:

2x + y + k = 0........................(2)

The point (5, 0) passes through the equation (2)

2x + y + k = 0

or, 2× 5 + 0 + k = 0

or, 10 + k = 0

∴ k = -10

Putting the value of k in equation (2),

2x + y - 10 = 0 Ans

Here,

Given equation is:

5x + 7y - 14 = 0.........................(1)

Equation (1) changes in perpendicular form

7x - 5y + k = 0............................(2)

Point (-2, -3) passes through the equation (2)

7× (-2) - 5× (-3) + k = 0

or, -14 + 15 + k = 0

or, 1 + k = 0

∴k = -1

Putting the value of k in equation (2)

7x - 5y - 1 = 0

∴ Required equation is:7x - 5y - 1 = 0 Ans

Given equation is:

4x - 3y = 10

i.e. 4x - 3y - 10 = 0..................................(1)

Equation (1) changes in the perpendicular form

3x + 4y + k = 0.........................................(2)

Point (2, 3) passes through the equation (2)

3× 2 + 4× 3 + k = 0

or, 6 + 12 + k = 0

or, 18 + k = 0

∴ k = -18

Substituting the value of k in equation (2)

3x + 4y - 18 = 0

∴ Required equation is: 3x + 4y - 18 = 0 Ans

Given equation is: 3x + 4y + 18 = 0.........................................(1)

Equation (1) changes in perpendicular form:

4x - 3y + k = 0.....................................(2)

Point (2, -3) passes through the equation (2)

4× 2 - 3× (-3) + k = 0

or, 8 + 9 + k = 0

or, 17 + k = 0

∴ k = -17

Putting the value of k in eqn (2)

4x - 3y - 17 = 0

∴ The required equation is:4x - 3y - 17 = 0 Ans

Given equation is: x - 2y - 2 = 0........................................(1)

Equation (1) changes in perpendicular form:

2x + y + k = 0.....................................(2)

Point (4, 6) passes through the equation (2)

2 × 4 + 6 + k = 0

or, 8 + 6+ k = 0

or, 14 + k = 0

∴ k = -14

Putting the value of k in eqn (2)

2x + y - 14 = 0

∴ The required equation is: 2x + y - 14 = 0Ans

Given equation is: x - 3y - 2 = 0............................(1)

Equation (1) changes in perpendicular form:

3x + y + k = 0.....................................(2)

Point (2, 3) passes through the equation (2)

3× 2+ 3+ k = 0

or, 6+ 3+ k = 0

or, 9+ k = 0

∴ k = -9

Putting the value of k in eqn (2)

3x + y -9 = 0

∴ The required equation is: 3x + y -9 = 0 Ans

Vertices of $$\triangle$$PQR are: P(-2, 1), Q(2, 3) and R(-2, -4).

We know that:

y - y1 = $$\frac {y_2 - y_1}{x_2 - x_1}$$(x- x1)

Eqn of QR,

y - 3 = $$\frac {-4 - 3}{-2 - 2}$$(x - 2)

or, y - 3 = $$\frac {-7}{-4}$$(x - 2)

or, y - 3 = $$\frac {7}{4}$$(x - 2)

or, 4y - 12 = 7x - 14

or, 7x - 14 - 4y +12 = 0

or, 7x - 4y - 2 = 0..................................(1)

The eqn (1) change in perpendicular form,

4x + 7y + k = 0.......................................(2)

The point (-2, 1) passes through eqn (1),

4× (-2) + 7× 1 + k = 0

or, -8 + 7 + k = 0

or, -1 + k = 0

∴ k = 1

Putting the value of k in eqn (2)

4x + 7y + 1 = 0 Ans

Given:

A(-1, 5), B(-4, -1) and C(3, -2) are the vertices of $$\triangle$$ABC.

We know,

Equation of the line passing through B(-4, -1) and C(3, -2) is:

y - y1 = $$\frac {y_2 - y_1}{x_2 - x_1}$$(x - x1)

or, y + 1 = $$\frac {-2 + 1}{3 + 4}$$(x + 4)

or, y + 1 = $$\frac {-1}7$$(x + 4)

or, 7 (y + 1) = -1 (x + 4)

or, 7y + 7 = -x - 4

or, x + 7y + 4 +7 = 0

∴ x + 7y + 11 = 0..................................(1)

The line parallel to the equation (1)

x + 7y + k = 0..........................................(2)

The equation (2) passes through the point A(-1, 5)

-1 + 7× 5 + k = 0

or, -1 + 35 + k = 0

or, 34 + k = 0

∴ k = -34

Putting the value of k in the equation (2)

x + 7y - 34 = 0

∴ The required equation is:x + 7y - 34 = 0 Ans

Here,

ABC is a triangle. The vertices of $$\triangle$$ABC are A(2, 8), B(-3, 5) and C(5, 3).

D is the mid-point of BC.

Co-ordinates of D

= ($$\frac {x_1 + x_2}2$$, $$\frac {y_1 + y_2}2$$)

= ($$\frac {-3 + 5}2$$, $$\frac {5 + 3}2$$)

= ($$\frac 22$$, $$\frac 82$$)

= (1, 4)

Slope of BC (m1)

= $$\frac {y_2 - y_1}{x_2 - x_1}$$

= $$\frac {3 - 5}{5 + 3}$$

= $$\frac {-2}8$$

= $$\frac {-1}4$$

Slope of AD (m2)

= $$\frac {y_2 - y_1}{x_2 - x_1}$$

= $$\frac {4 - 8}{1 - 2}$$

= $$\frac {-4}{-1}$$

= 4

Now,

m1m2 = $$\frac {-1}4$$× 4 = -1

The product of two slope is equal to -1 so these are perpendicular.

∴ AD⊥ BC Proved

Again,

Equation of AD;

y - y1 =$$\frac {y_2 - y_1}{x_2 - x_1}$$(x - x1)

or, y - 8 = $$\frac {4 - 8}{1 - 2}$$(x - 2)

or, y - 8 = $$\frac {-4}{-1}$$(x - 2)

or, y - 8 = 4x - 8

or, 4x - 8 - y + 8 = 0

∴ 4x - y = 0

Hence, the required equation is:4x - y = 0 Ans

Here,

The vertices of $$\triangle$$ABC are: A(3, 4), B(-2, 2) and C(3, -3).

Equation of BC is:

y - y1= $$\frac {y_2 - y_1}{x_2 - x_1}$$(x - x1)

or, y - 2 = $$\frac {-3 - 2}{3 + 2}$$(x + 2)

or, y - 2 = $$\frac {-5}5$$(x + 2)

or, y - 2 = -x - 2

or, x + 2 + y - 2 = 0

∴ x + y = 0.......................................(1)

The eqn (1) changes in perpendicular form,

x - y + k = 0......................................(2)

The point A(3, 4) passes through eqn (1)

3 - 4 + k = 0

or, -1 + k = 0

∴ k = 1

Putting the value of k in eqn (1)

x - y + 1 = 0

∴ The required equation is:x - y + 1 = 0 Ans

Given equation is: 7x - 24y + 10 = 0......................(1)

Equation (1) change in perpendicular form

24x + 7y + k = 0..............................(2)

Equation (2) passes through the point (-2, 4)

24× (-2) + 7× 4 + k = 0

or, -48 + 28 + k = 0

or, -20 + k = 0

∴ k = 20

Putting the value of k in equation (2)

24x + 7y + 20 = 0

Equation of the line PQ is: 24x + 7y + 20 = 0

We know,

d = $$\begin {vmatrix} \frac {Ax + By + C}{\sqrt {A^2 + B^2}}\\ \end {vmatrix}$$

Perpendicular length of PQ;

=$$\begin {vmatrix} \frac {7 × (-2) - 24 × 4 + 10}{\sqrt {7^2 + (24)^2}}\\ \end {vmatrix}$$

=$$\begin {vmatrix} \frac {-14 - 96 + 10}{\sqrt {49 + 576}}\\ \end {vmatrix}$$

=$$\begin {vmatrix} \frac {-100}{\sqrt {625}}\\ \end {vmatrix}$$

=$$\begin {vmatrix} \frac {-100}{25}\\ \end {vmatrix}$$

=$$\begin {vmatrix} -4\\ \end {vmatrix}$$

= 4 units

∴ The perpendicular length of PQ = 4 units and equation of PQ is:24x + 7y + 20 = 0 Ans

Let: m be the slope of the required line, so that its equation is:

y - y1 = m(x - x1).............................(1)

Point (2, -1) passes through equation (1):

y + 1 = m(x - 2)...............................(2)

Given equation is:

6x + 5y - 1 = 0.................................(3)

Slope of eqn (1) is:

slope (m1) = - $$\frac {x-coefficient}{y-coefficient}$$ = - $$\frac 65$$

Now,

Using angle formula,

tan$$\theta$$ =± ($$\frac {m_1 - m_2}{1 + m_1m_2}$$)

or, tan 45° =± ($$\frac {m + \frac 65}{1 - m\frac 65}$$)

or, 1 =± ($$\frac {\frac {5m + 6}5}{\frac {5 - 6m}5}$$)

or, 1 =± ($$\frac {5m + 6}{5 - 6m}$$)

Taking +ve sign,

5 - 6m = 5m + 6

or, -6m - 5m = 6 - 5

or, -11m = 1

∴ m = -$$\frac 1{11}$$

Taking -ve sign,

5 - 6m = - (5m + 6)

or, 5 - 6m = -5m - 6

or, -6m + 5m = - 6 - 5

or, - m = - 11

∴ m = 11

Putting the value of m = -$$\frac 1{11}$$ in equation (2)

y + 1 = m (x - 2)

or, y + 1 = -$$\frac 1{11}$$(x - 2)

or, 11y + 11 = -x + 2

or, x + 11y + 11 - 2 = 0

∴ x + 11y + 9 = 0

Putting the value of m = 11 in equation (2)

y + 1 = m(x - 2)

or, y + 1= 11 (x - 2)

or, y + 1 = 11x - 22

or, 11x - y - 22 - 1 = 0

∴ 1 1x - y - 23 = 0

Hence, the required equations are: x + 11y + 9 = 0 and 11x - y - 23 = 0 Ans

Given:

The eqn of AB is:

12(x + 3) = 5y

or, 12x + 36 - 5y = 0

or, 12x - 5y + 36 = 0........................(1)

Slope of eqn (1) is: m1 = $$\frac {12}5$$

The eqn of the line passes through (2, 3) is:

y - y1 = m (x - x1)

or, y - 3 = m (x - 2)..........................(2)

Slope of eqn (2) is: m2 = m

The angle between the lines (1) and (2) is 45°.

tan$$\theta$$ =± $$\frac {m_1 - m_2}{1 + m_1m_2}$$

or, tan 45° =± $$\frac {\frac {15}2 - m}{1 + \frac {12}5m}$$

or, 1 =± $$\frac {\frac {12 - 5m}5}{\frac {5 + 12m}5}$$

or, 1 =± $$\frac {12 - 5m}{5 + 12m}$$

Taking +ve sign,

5 + 12m = 12 - 5m

or, 12m + 5m = 12 - 5

or, 17m = 7

∴ m = $$\frac 7{17}$$

Taking -ve sign,

5 + 12m = - 12 + m

or, 12m - 5m = - 12 - 5

or, 7m = - 17

∴ m = -$$\frac {17}7$$

Substituting the value of m = $$\frac 7{17}$$ in eqn (2)

y - 3 = $$\frac 7{17}$$(x - 2)

or, 17(y - 3) = 7(x - 2)

or, 17y - 51 = 7x - 14

or, 7x - 17y + 51 - 14 = 0

∴ 7x - 17y + 37 = 0

Substituting the value of m = $$\frac {-17}7$$ in eqn (2)

y - 3 = $$\frac {-17}7$$(x - 2)

or, 7 (y - 3) = -17 (x - 2)

or, 7y - 21 = - 17x + 34

or, 17x - 34 + 7y - 21 = 0

∴ 17x + 7y - 55 = 0

∴ The required equations are:7x - 17y + 37 = 0 and17x + 7y - 55 = 0 Ans

Here,

The eqn of line is:

$$\sqrt 3$$x - y = 4.............................(1)

The slope of eqn (1) is: m1 =$$\sqrt 3$$

The eqn of the line passes through (4, 3) is:

y - 3 = m(x - 4)..........................(2)

Slope of eqn (2) is m2 = m

The angle between eqn (1) and (2) is 60°.

tan$$\theta$$ =± $$\frac {m_1 - m_2}{1 + m_1m_2}$$

or, tan 60° =± $$\frac {\sqrt 3 -m}{1 + \sqrt3m}$$

or, $$\sqrt 3$$ =± $$\frac {\sqrt 3 - m}{1 + \sqrt 3m}$$

Taking +ve sign,

$$\sqrt 3$$(1 + $$\sqrt 3$$m) = $$\sqrt 3$$ - m

or, $$\sqrt 3$$ + 3m = $$\sqrt 3$$ - m

or, 3m + m = $$\sqrt 3$$ - $$\sqrt3$$

or, 4m = 0

∴ m = 0

Taking -ve sign,

$$\sqrt 3$$ + 3m = -$$\sqrt 3$$ + m

or, 3m - m = -$$\sqrt 3$$ - $$\sqrt 3$$

or, 2m = - 2$$\sqrt 3$$

or, m = $$\frac {-2\sqrt 3}2$$

∴ m = -$$\sqrt 3$$

Putting the value of m = 0 in eqn (2)

y - 3 = m(x - 4)

or, y - 3 = 0(x - 4)

∴ y - 3 = 0

Putting the value of m = -$$\sqrt 3$$ in eqn (2)

y - 3 = m (x - 4)

or, y - 3 = -$$\sqrt 3$$ (x - 4)

or, y - 3 = -$$\sqrt 3$$x + 4$$\sqrt 3$$

∴ $$\sqrt 3$$x + y = 4$$\sqrt 3$$ + 3

∴ The required equations are: y - 3 = 0 and $$\sqrt 3$$x + y = 4$$\sqrt 3$$ + 3 Ans

Here,

The given eqn is:

2x + 3y + 5 = 0............................(1)

Slope of equation (1) is: m1 = $$\frac {-2}3$$

The equation of the line passes through (1, -4) is:

y + 4 = m(x - 1)...........................(2)

Slope of equation (2) is: m2 = m

The angle between the lines (1) and (2) is 45°.

tan$$\theta$$ =± ($$\frac {m_1 - m_2}{1 + m_1m_2}$$)

or, tan 45° =± ($$\frac {\frac {-2}3 - m}{1 + m(\frac {-2}3)})$$

or, 1 =± ($$\frac {\frac {-2 - 3m}3}{\frac {3 - 2m}3})$$

or, 1 =± ($$\frac {-2 - 3m}{3 - 2m})$$

Taking +ve sign,

1 = $$\frac {-2 - 3m}{3 - 2m}$$

or, -2 - 3m = 3 - 2m

or, -3m + 2m = 3 + 2

or, -m = 5

∴ m = -5

Taking -vesign,

1 = $$\frac {-(-2 - 3m)}{3 - 2m}$$

or, 3 - 2m = 2 + 3m

or, 3m + 2m = 3 - 2

or, 5m = 1

∴ m = $$\frac 15$$

∴ m = -5, $$\frac 15$$

Substituting the value of m = -5 in equation (2)

y + 4 = - 5 (x - 1)

or, y + 4 = - 5x + 5

or, 5x + y + 4 - 5 = 0

∴ 5x + y - 1 = 0

Substituting the value of m = $$\frac 15$$ in equation (2)

y + 4 = $$\frac 15$$(x - 1)

or, 5y + 20 = x - 1

or, x - 1 - 5y - 20 = 0

∴ x - 5y - 21 = 0

Hence, the required equations are:5x + y - 1 = 0 andx - 5y - 21 = 0 Ans

Here,

The given equation is:

x - 3y = 2.............................(1)

The slope of eqn (1) is: m1 = -$$\frac {x-coefficient}{y-coefficient}$$ = $$\frac {-1}{-3}$$ = $$\frac 13$$

Let, the equation of the line passes through point (2, 3) is;

y - 3 = m(x - 2)....................(2) [$$\because$$y - y1 = m(x - x1)]

The slope of equation (2) is: m2 = m

If the angle between the lines (1) and (2) is 45°, then:

tan$$\theta$$ =± ($$\frac {m_1 - m_2}{1 + m_1m_2})$$

or, tan 45° =± ($$\frac {\frac 13 - m}{1 + \frac 13m})$$

or, 1 =± $$\frac {\frac {1 - 3m}3}{\frac {3 + m}3}$$

or, 1 =± $$\frac {1 - 3m}{3 + m}$$

Taking +ve sign,

1 = $$\frac {1 - 3m}{3 + m}$$

3 + m = 1 - 3m

or, 3m + m = 1 - 3

or, 4m = - 2

or, m = -$$\frac 24$$

∴ m = - $$\frac 12$$

Taking -ve sign,

1 = -$$\frac {1 - 3m}{3 + m}$$

or, 3 + m = - (1 - 3m)

or, 3 + m = -1 + 3m

or, 3m - m = 3 + 1

or, 2m = 4

or, m = $$\frac 42$$

∴ m = 2

Putting the value of m = - $$\frac 12$$ in eqn (2)

y - 3 = -$$\frac 12$$(x - 2)

or, 2y - 6 = - x + 2

or, x + 2y - 2 - 6 = 0

∴ x + 2y - 8 = 0

Putting the value of m = 2 in eqn (2)

y - 3 = 2(x - 2)

or, y - 3 = 2x - 4

or, 2x - 4 - y + 3 = 0

∴ 2x - y - 1 = 0

∴ The required equations are: x + 2y - 8 = 0 and2x - y - 1 = 0 Ans

Let: AB and AC be two lines whose equations are: y = m1x + c1 and y = m2x + c2, these lines meet OX in the point B and C respectively.

Let: $$\angle$$ABX = $$\theta_1$$ and $$\angle$$ = $$\theta_2$$

Let: tan$$\theta_1$$ = m1 and tan$$\theta_2$$ = m2

Let: $$\angle$$CAB = $$\theta$$, then;

$$\theta_1$$ = $$\theta$$ + $$\theta_2$$

∴ $$\theta$$ = $$\theta_1$$ - $$\theta_2$$

Taking tan on both sides,

tan$$\theta$$ = tan($$\theta_1$$ - $$\theta_2$$)

or, tan$$\theta$$ = $$\frac {tan\theta_1 - tan\theta_2}{1 + tan\theta_1\theta_2}$$

or, tan$$\theta$$ = $$\frac {m_1 - m_2}{1 + m_1m_2}$$

∴ $$\theta$$ = tan-1($$\frac {m_1 - m_2}{1 + m_1m_2}$$) Ans

Given equations are:

a1x + b1y + c1 = 0............................(1)

a2x + b2y + c2 = 0............................(2)

Let: m1 and m2 represents the slope of the lines (1) and (2) respectively.

Slope of (1), m1= -$$\frac {x-coefficient}{y-coefficient}$$ = -$$\frac {a_1}{b_1}$$

Slope of (2), m2= -$$\frac {x-coefficient}{y-coefficient}$$ = -$$\frac {a_2}{b_2}$$

Let: $$\theta$$ be the angle between the two given lines. Then,

tan$$\theta$$ =± ($$\frac {m_1 - m_2}{1 + m_1m_2})$$

or, tan$$\theta$$ =± ($$\frac {(\frac {-a_1}{b_1}) - (\frac {-a_2}{b_2})}{1 + (\frac {-a_1}{b_1})(\frac {-a_2}{b_2})})$$

or, tan$$\theta$$ =± ($$\frac {\frac {-a_1b_2 + a_2b_1}{b_1b_2}}{\frac {b_1b_2 + a_1a_2}{b_1b_2}})$$

or, tan$$\theta$$ =± ($$\frac {a_2b_1 - a_1b_2}{a_1a_2 + b_1b_2})$$

∴ $$\theta$$ = tan-1± ($$\frac {a_2b_1 - a_1b_2}{a_1a_2 + b_1b_2})$$ Ans

Let: ABC is an equilateral triangle with base BC on x-axis.

i.e. y = 0, $$\angle$$CBA = 60° and $$\angle$$XCA = 120°

Slope of line AB = m1 = tan$$\theta$$ = tan 60° = $$\sqrt 3$$

Slope of line CA = m2 = tan$$\theta$$ = tan 120° = - $$\sqrt 3$$

The eqn of AB is:

y - y1 = m(x - x1)

or, y - 2 = $$\sqrt 3$$(x + 1)

or, y - 2 = $$\sqrt 3$$x + $$\sqrt 3$$

∴$$\sqrt 3$$x - y + 2 + $$\sqrt 3$$ = 0

The eqn of the sides CA is:

y - y1 = m(x - x1)

or, y - 2 = -$$\sqrt 3$$ (x + 1)

or, y - 2 = -$$\sqrt 3$$x - $$\sqrt 3$$

∴$$\sqrt 3$$x + y - 2 + $$\sqrt 3$$ = 0

Hence, the required lines are:$$\sqrt 3$$x - y + 2 + $$\sqrt 3$$ = 0 and$$\sqrt 3$$x + y - 2 + $$\sqrt 3$$ = 0 Ans

Given equations are:

x + 2y = 3...........................(1)

2x - 3y = 20......................(2)

Eqn (1) is multipliedby 2 and subtract with eqn (2)

 2x + 4y = 6 2x - 3y = 20 - + - 7y = -14

y = -$$\frac {14}7$$

∴ y = -2

Putting the value of y in eqn (1)

x + 2y = 3

or, x + 2× (-2) = 3

or, x - 4 = 3

or, x = 3 + 4

∴ x = 7

The point of intersection is: (7, -2)

The given eqn is:

2x - 3y + 5 = 0.............................(3)

The eqn (3) changes in perpendicular form:

-3x - 2y + k = 0..........................(4)

The point (7, -2) passes through eqn (4)

-3× 7 - 2× (-2) + k = 0

or, -21 + 4 + k = 0

or, -17 + k = 0

∴ k = 17

Putting the value of k in eqn (4)

-3x - 2y + 17 = 0

∴ 3x + 2y - 17 = 0 Ans

Given points are: (-3, -4) and (7, 1)

Eqn of two points is:

y - y1 = $$\frac {y_2 - y_1}{x_2 - x_1}$$(x - x1)

or, y + 4 = $$\frac {1 + 4}{7 + 3}$$(x + 3)

or, y +4 = $$\frac 5{10}$$(x + 3)

or, y +4 = $$\frac 12$$(x + 3)

or, 2y + 8 = x + 3

or, x + 3 - 2y - 8 = 0

∴ x - 2y - 5 = 0.............................(1)

The points (-3, -4) and (7, 1) divides by a point P(x, y) in the ratio 3 : 2.

Using section formula,

P(x, y) = ($$\frac {m_1x_2 + m_2x_1}{m_1 + m_2}$$, $$\frac {m_1y_2 + m_2y_1}{m_1 + m_2})$$

or,P(x, y) = ($$\frac {3 × 7 + 2 × (-3)}{3 + 2}$$, $$\frac {3 × 1 + 2 × (-4)}{3 + 2}$$)

or,P(x, y) = ($$\frac {21 - 6}5$$, $$\frac {3 - 8}5$$)

or,P(x, y) = ($$\frac {15}5$$, $$\frac {-5}5$$)

∴ P(x, y) = (3, -1)

The eqn (1) changes in perpendicular form is:

-2x - y + k = 0....................................(2)

The point (3, -1) passes through eqn (2)

-2× 3 - (-1) + k = 0

or, -6 + 1 + k = 0

or, -5 + k = 0

∴ k = 5

Putting the value of k in eqn (2)

-2x - y + 5 = 0

∴ 2x + y - 5 = 0 Ans

Here,

The diagonals of the square bisect each other.

Mid-point of AC = ($$\frac {x_1 + x_2}2$$, $$\frac {y_1 + y_2}2$$) = ($$\frac {2 - 6}2$$, $$\frac {3 + 5}2$$) = ($$\frac {-4}2$$, $$\frac 82$$) = (-2, 4)

The eqn of the points (2, 3) and (-6, 5) is:

y - y1 = $$\frac {y_2 - y_1}{x_2 - x_1}$$(x - x1)

or, y - 3 = $$\frac {5 - 3}{-6 - 2}$$(x - 2)

or, y - 3 = $$\frac 2{-8}$$(x - 2)

or, y - 3 = $$\frac 1{-4}$$(x - 2)

or, - 4y + 12 = x - 2

or, x - 2 + 4y - 12 = 0

∴ x + 4y - 14 = 0.............................(1)

Since, the diagonals of the square are perpendicular to each other.

The eqn (1) change in perpendicular form:

4x - y + k = 0.....................................(2)

The point (-2, 4) passes through eqn (2)

4× (-2) - 4 + k = 0

or, -8 - 4 + k = 0

or, - 12 + k = 0

∴ k = 12

Putting the value of k in eqn (2)

4x - y + 12 = 0

∴ The required equation is:4x - y + 12 = 0 Ans