math help thank you

Question

Find the points on the ellipse 4x^2+y^2=4 that are farthest from the point (-1,0)

please be detailed thank you!

Answer 1

Finding the farthest point from (-1,0) means that we’re trying to maximize the distance between the ellipse and the point (-1,0)

The distance formula is D = sqrt((x₁ - x₂)² + (y₁ - y₂)²)

Plug in the point (-1,0)

D = sqrt((x₁ + 1)² + (y₁ - 0)²)

D = sqrt((x₁ + 1)² + (y₁)²)

(x₁, y₁) is a point on the ellipse, thus we can just use variables x and y.

D = sqrt((x + 1)² + y²)

Get everything in terms of x.

y² = 4 - 4x²

So substitute y² in our distance equation with 4 - 4x²

D = sqrt((x + 1)² + 4 - 4x²)

Maximize D