Question

The perpendicular from the origin to a line meets it at the point (– 2, 9), find the equation of the line.

Answer 1

The perpendicular is passing through ( 0,0 ) and meets the line at ( −2,9 ) .

The formula for the slope of a line passing through two different 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 passing through point ( 0,0 ) and ( −2,9 ) .

Substitute the values of ( x 1 , y 1 ) and ( x 2 , y 2 ) as ( 0,0 ) and ( −2,9 )

m 1 = 9−0 −2−0 =− 9 2

The product of the slopes of perpendicular lines is equal to -1.

m 1 ⋅ m 2 =−1 (2)

Here m 2 be the slope of line segment which is perpendicular to the line passing through point ( 0,0 ) and ( −2,9 ) .

Substitute the value of m 1 in equation (2).

− 9 2 ⋅ m 2 =−1 m 2 = −1 −9 2 = 2 9

Now, 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 ) (3)

Substitute the value of ( x 0 , y 0 ) as ( −2,9 ) and slope m as 2 9 in equation (3).

( y−9 )= 2 9 ⋅( x−( −2 ) ) 9⋅( y−9 )=2⋅( x+2 ) 9y−81=2x+4 2x−9y+85=0

Thus, the equation of line passing through point ( −2,9 ) is 2x−9y+85=0 .