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Find p and q if the equation px2 – 8xy + 3y2 + 14x + 2y + q = 0 represents a pair of perpendicular lines.

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

∴ (coefficient of x2) + (coefficient of y2) = 0

∴ p + 3 = 0 p = -3

With this value of p, the given equation is

– 3x2 – 8xy + 3y2 + 14x + 2y + q = 0.

Comparing this equation with

ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, we have,

a = -3, h = -4, b = 3, g = 7, f = 1 and c = q.

= -3(3q – 1) + 4(-4q – 7) + 7(-4 – 21)

= -9q + 3 – 16q – 28 – 175

= -25q – 200= -25(q + 8)

Since the given equation represents a pair of lines, D = 0

∴ -25(q + 8) = 0 ∴ q= -8.

Hence, p = -3 and q = -8.

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