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

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

#### Solution

Given equation is px2 - 8xy + 3y2 + 14x + 2y + q = 0 Comparing with ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, we get

a = p, h = - 4, b = 3, g = 7, f = 1, c = q.
The given equation represents a pair of lines perpendicular to each other
∴ a + b = 0
∴ p + 3 = 0
∴ p = -3
Also, the given equation represents a pair of lines

[[a,h,g],[h,b,f],[g,f,c]]=0

|[-3,-4,7],[-4,3,1],[7,1,q]|=0

∴3(3q - 1) + 4( - 4q - 7) + 7(- 4 - 21) = 0

∴-9q + 3 - 16q -28 - 175 = 0

∴-25q-200=0

∴-25q=200

∴q=-8

∴p=-3 and q=-8

Concept: Concept of Line - Equation of Line Passing Through Given Point and Parallel to Given Vector
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