abx² + 2rxy - (k+b)y² = 0 | Find the value of k | Pair Of Straight Lines | SciPiPupil

Question: Find the value of k if the lines represented by the following equations are perpendicular to each other: abx² + 2rxy - (k+b)y² = 0.

Solution:

Single equation of straight lines is $abx² + 2rxy - (k+b)y²$ - (i)

Comparing equation (i) with $ax² + 2hxy + by² = 0$

$a = ab \; and \; b = -(k+b)$

Given, two lines represented by the above equation are perpendicular to each other then,

$a + b= 0$

$or, ab - (k+b) = 0$

$or, ab - k - b = 0$

$or, ab - b = k$

$or, k = ab - b$

$\therefore k = b(a -1)$

Hence, the required value of r in the above equation is b(a -1).