Show that the following curves intersect orthogonally at the indicated points :
y² = 8x and 2x² + y² = 10 at (1, 2√2)

Question

Show that the following curves intersect orthogonally at the indicated points : y 2 = 8x and 2x 2 + y 2 = 10 at (1, 2√2)

Philoid · Accepted Answer

Given:

Curves y² = 8x ...(1)

& 2x² + y² = 10 ...(2)

The point of intersection of two curves are (0,0) & (1,2)

Now ,Differentiating curves (1) & (2) w.r.t x, we get

⇒ y² = 8x

⇒ 2y.8

⇒

⇒ ...(3)

⇒ 2x² + y² = 10

Differentiating above w.r.t x,

⇒ 4x + 2y. = 0

⇒ 2x + y. = 0

⇒ y. = – 2x

...(4)

Substituting (1,2)for m₁ & m_2,we get,

m₁

⇒

m₁ = ...(5)

m₂

⇒

m₂ = ...(6)

when m₁ & m₂

⇒ ×1

∴ Two curves y² = 8x & 2x² + y² = 10 intersect orthogonally.

Show that the following curves intersect orthogonally at the indicated points :y2 = 8x and 2x2 + y2 = 10 at (1, 2√2)