Question

If equation ax² - y² + 2y + c = 1 represents a pair of perpendicular lines, then find a and c.

Answer 1

The given equation represents a pair of lines perpendicular to each other.

∴ coefficient of x² + coefficient of y² = 0

∴ a - 1 = 0

∴ a = 1

With this value of a, the given equation is

x² - y² + 2y + c - 1 = 0

Comparing this equation with

Ax² + 2Hxy + By² + 2Gx + 2Fy + C = 0, we get,

A = 1, H = 0, B = -1, G = 0, F = 1, C = c - 1

Since the given equation represents a pair of lines,

D = `|("A","H","G"),("H","B","F"),("G","F","C")| = 0`

∴ `|(1,0,0),(0,-1,1),(0,1,"c - 1")| = 0`

∴ 1(- c + 1 - 1) - 0 + 0 = 0

∴ - c = 0

∴ c = 0

Hence, a = 1, c = 0.