Question

If the equation 12x²– 10xy + 2y² + 14x – 5y + c = 0 represents a pair of straight lines, find the value of c. Find the separate equations of the straight lines and also the angle between them.

Answer 1

12x² – 10xy – 2y² + 14x – 5y + c = 0

ax²+ 2hxy + by² + 2gx + 2fy – c = 0 

Here a = 12, b = 2, g = 7, 

f = 5/2, c = c, h = -5 

af² + bg² + ch² – 2fgh – abc = 0 is the condition

The equation is 12x² – 10y + 2y² + 14x – 5y + 2 = 0 

12x² – 10xy + 2y = (3x – y)(4x – 2y) 

Let 12x² – 10y + 2y² + 14x – 5y + 2(3x – y + l)(4x – 2y + m)

So that 4l + 3m = 14 , -2l – m = -5

On solving we get l = 1/2, m = 4