Question

The line through the points ( h , 3) and (4, 1) intersects the line 7 x – 9 y – 19 = 0 . at right angle. Find the value of h .

Answer 1

The line passes through the points ( h,3 ) and ( 4,1 ) . Also this line intersects the line 7x−9y−19=0 at right angles.

The equation of line having slope m and making an intercept c with the y axis is given by

y=mx+c (1)

Let m 1 is the slope of the line 7x−9y−19=0 .

Rearrange the terms of the equation of line 7x−9y−19=0 .

−9y=−( 7x−19 ) 9y=7x−19 y= 7 9 x− 19 9

Compare the above equation with the equation (1).

m 1 = 7 9

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 (2)

Let m 2 be the slope of the line passes through the points ( h,3 ) and ( 4,1 ) .

Substitute the values of ( x 1 , y 1 ) , ( x 2 , y 2 ) as ( h,3 ) and ( 4,1 ) in equation (2).

m 2 = 1−3 4−h = −2 4−h

If two lines are perpendicular to each other the product of their slopes is equal to −1 .

m 1 ⋅ m 2 =−1

Substitute the values of m 1 and m 2 in above expression.

( 7 9 )×( −2 4−h )=−1 −14 9×( 4−h ) =−1 −14=−1×9×( 4−h ) 14=9×( 4−h )

Further simplify the above equation.

14=36−9h 9h=36−14 9h=22 h= 22 9

Thus, the value of h is 22 9 .