Q. Find the angle between the lines represented by the equation (x^2 + y^2)sin^2a = (xcosa – ysina)^2.

Soln:

(x^2 + y^2)sin²a = (xcosa – ysina)^2

x²sin²a + y²sin²a = x²cos²a – 2xycosa.sina + y²sin²a

(sin²a – cos²a)x² + 2xysina.cosa = 0

Comparing with ax² + 2hxy +by² = 0

a = sin²a – cos²a and h = sina.cosa and c = 0

If  q be the angle between the lines represented by eqn(i), then

tanq = (plus/minus) 2[Ö(h² – ab)]/(a + b)

tanq = (plus/minus) 2Ö[(sina.cosa)² – (sin²a – cos²a).0]/(sin²a – cos²a + 0) 

tanq = (plus/minus) 2 [Ö(sin²a.cos²a)]/(sin²a – cos²a)    

tanq = (plus/minus) [(2sina.cosa)/(cos²a – sin²a)]   [Taking -ve sign common]

tanq = (plus/minus) [(sin2a/cos2a)]  [sin2a = 2sina.cosa; cos2a = sin²a – cos²a]

tanq = (plus/minus) tan2a

q = (plus/minus) 2a

Note: In 4th last step, negative sign has been taken common and it will change plus into minus and minus into plus sign, however there will not be any change in (plus/minus).