If the straight lines ax + by + p = 0 and x cosα + y sinα = p are inclined at an angle…

Problem: If the straight lines ax + by + p = 0 and x cosα + y sinα = p are inclined at an angle π/4 and concurrent with the straight line x sinα – y cosα = 0, then the value of a² + b² is

(A) 0

(B) 1

(C) 2

(D) none of these

Sol:

ON = p( see fig.)

OM = perpendicular distance of O from ax + by + p =0

$\large = \frac{p}{\sqrt{a^2 + b^2}}$

$\large = ON sin\pi/4 = \frac{p}{\sqrt{2}}$

Hence , a² + b² = 2