Question

Find points on the ellipse x²+2y²=9 at which tangent has slope 1/4.

Answer 1

we have to find the points on the ellipse, x² + 2y² = 9 at which tangent has slope 1/4.

first, differentiate equation of ellipse with respect to x,

i.e., 2x + 4y dy/dx = 0

⇒dy/dx = -x/2y

hence, slope of tangent of the ellipse is , dy/dx = -x/2y

but given, dy/dx = 1/4

so, -x/2y = 1/4

⇒y = -2x putting in equation of ellipse.

so, x² + 2(-2x)² = 9

⇒x² + 8x² = 9

⇒9x² = 9 ⇒x = ±1

y = -2x = $\mp 2$

hence, points are (1,-2) and (-1, 2)