Question

Find the equation of the line passing through (–3, 5) and perpendicular to the line through the points (2, 5) and (–3, 6).

Answer 1

The line passes through point ( −3,5 ) and perpendicular to the line passes through the points ( 2,5 ) and ( −3,6 ) .

The formula for the slope of a line passes through points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by,

m= y 2 − y 1 x 2 − x 1 (1)

Let m 1 be the slope of the line which passes through the points ( 2,5 ) and ( −3,6 )

Substitute the value for ( x 1 , y 1 ) and ( x 2 , y 2 ) as ( 2,5 ) and ( −3,6 ) respectively in equation (1).

m 1 = 6−5 −3−2 = 1 −5 =− 1 5 (2)

Since the line which passes the point ( −3,5 ) is perpendicular to the line passing through the points ( 2,5 ) and ( −3,6 ) , the product for slopes of these two perpendicular lines are equal to −1 .

m 1 ⋅ m 2 =−1 (3)

Where m 2 be the slope of the perpendicular line through point ( −3,5 ) .

Substitute the value of m 1 from equation (2) to equation (3).

− 1 5 ⋅ m 2 =−1 m 2 = −1 ( −1 5 ) =5

The formula for the equation of a non-vertical line having slope m and passing through the point ( x 0 , y 0 ) is given by,

( y− y 0 )=m( x− x 0 ) (4)

Substitute the value of ( x 0 , y 0 ) as ( −3,5 ) and m as 5.in equation (4).

( y−5 )=5( x−( −3 ) ) ( y−5 )=5( x+3 ) y−5=5x+15 5x−y+15+5=0

Rearrange the terms in above equation

5x−y+20=0

Thus, the equation of line passes through point ( −3,5 ) and also perpendicular to the line passes through the points ( 2,5 ) and ( −3,6 ) is 5x−y+20=0 .