Find the angle to intersection of the following curves :
x2 + 4y2 = 8 and x2 – 2y2 = 2
Given:
Curves x2 + 4y2 = 8 ...(1)
& x2 – 2y2 = 2 ...(2)
Solving (1) & (2),we get,
from 2nd curve,
x2 = 2 + 2y2
Substituting on x2 + 4y2 = 8,
⇒ 2 + 2y2 + 4y2 = 8
⇒ 6y2 = 6
⇒ y2 = 1
⇒ y = ±1
Substituting on y = ±1,we get,
⇒ x2 = 2 + 2(±1)2
⇒ x2 = 4
⇒ x = ±2
∴ The point of intersection of two curves (2,1) & ( – 2, – 1)
Now ,Differentiating curves (1) & (2) w.r.t x, we get
⇒ x2 + 4y2 = 8
⇒ 2x + 8y. = 0
⇒ 8y. = – 2x
...(3)
⇒ x2 – 2y2 = 2
⇒ 2x – 4y.0
⇒ x – 2y.0
⇒ 4yx
⇒ ...(4)
At (2,1) in equation(3),we get
m1
At (2,1) in equation(4),we get
= 1
m2 = 1
when m1 & m2 = 1
tanθ
tanθ
tanθ
tanθ
θ = tan – 1(3)
θ≅71.56