Question

The points where the normals to the ellipse x² + 3y² = 37 are parallel to the line 6x – 5y = 2 are ______.

Options

• (4, 2)(–5, –2)

• (5, 2)(–5, –3)

• (5, 2)(–5, –2)

• (5, –2)(–5, 2)

Answer 1

The points where the normals to the ellipse x² + 3y² = 37 are parallel to the line 6x – 5y = 2 are (5, 2)(–5, –2).

Explanation:

Given ellipse is x² + 3y² = 37

On differentiation `2x + 6y (dy)/(dx)` = 0

⇒ `(dy)/(dx) = (-2x)/(6y) = -x/(3y)` = slope of tangent

∴ Slope of normal = `(3y)/x`

Which is parallel to the line 6x – 5y = 2

∴ `(3y)/x = 6/5`

⇒ 6x – 15y = 0

⇒ 2x – 5y = 0

⇒ `x/y = 5/2`