What will be the value of angle between the curves x2 – y2 = 2a2 and xv + y2 = 4a2?

What will be the value of angle between the curves x2 – y2 = 2a2 and xv + y2 = 4a2? Correct Answer π/3

x2 – y2 = 2a2 ……….(1) and x2 + y2 = 4a2 ……….(2) Adding (1) and (2) we get, 2x2 = 6a2 Again, (2) – (1) gives, 2y2 = 2a2 Therefore, 2x2 * 2y2 = 6a2 * 2a2 4x2y2 = 12a2 Or x2y2 = 3a4 Or 2xy = ±2√3 Differentiating both side of (1) and (2) with respect to x we get, 2x – 2y(dy/dx) = 0 Or dy/dx = x/y And 2x + 2y(dy/dx) = 0 Ordy/dx = -x/y Let (x, y) be the point of intersection of the curves(1) and (2) and m1 and m2 be the slopes of the tangents to the curves (1) and (2) respectively at the point (x, y); then, m1 = x/y and m2 = -x/y Now the angle between the curves (1) and (2) means the angle between the tangents to the curve at their point of intersection. Therefore, if θ is the required angle between the curves (1) and (2), then tanθ = |(m1 – m2)/(1 + m1m2)| Putting the value of m1, m2 in the above equation we get, tanθ = |2xy/(y2 – x2)| As, 2xy = ±2√3a2 and x2 – y2 = 2a2 tanθ = |±2√3a2/-2a2| Or tanθ = √3 Thus, θ = π/3.