Question

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

Answer 1

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`