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Question:

Find the equations of the tangent and the normal to the curve 16x² + y² = 145 at the point (x₁, y₁), where x₁ = 2 and y₁ > 0.

Solution:

On differentiating the curve w.r.t x, 32x + 2y(dy/dx) = 0 ⟹ dy/dx(x₁, y₁) = -16x₁/y₁ ∴ tangent at (x₁, y₁) would be y - y₁ / x - x₁ = -16x₁/y₁ ⟹ y - y₁ = -16x₁/y₁(x - x₁) ⟹ yy₁ + 16x₁x = y₁² + 16x₁² ∴ tangent at (x₁, y₁) would be y - y₁ / x - x₁ = -16x₁/y₁ ⟹ y - y₁ = -16y₁/32(x - x₁) ⟹ 32y - 32y₁ = -16y₁(x - x₁) ⟹ y₁x + 32y = 16y₁ +32y₁. These are the required solutions.