How do you do this, Please show how

Question

Find the points on the ellipse 4𝑥²+𝑦² = 4 that are farthest away from the point (1,0).

Answer 1

Please note that there is an error in this solution. When you apply the distance formula, you should get:

d = sqrt[(x₀ - x)² + (y₀ - y)²] = sqrt[(1 - x)² + (0 - y)²] = sqrt[(1 - x)² + y²] = sqrt[(1 - x)² + 4 - 4x²].

Letting f(x) = d², we get:

f(x)= (1 - x)² + 4 - 4x² = 1 - 2x + x² + 4 - 4x² = -3x² - 2x + 5.

And then continue as Lilla H does.

Answer 2

Ethan,

We are looking for a point P(x,y) that is a point on the ellipse and its distance from (1,0) is the largest possible.

Because it is on the ellipse, its coordinates must satisfy the equation of the ellipse, so y=+- sqr(4-4x^2)

Apply the distance formula for these two points:

d = sqr[(1-x)^2 + (4-4x)^2]

We need to find the maximum of this expression. Since the largest the number the largest its square root, we can look for the maximum value of the function:

f(x)=1-x)^2 + (4-4x)^2 = 1 - 2x + x^2 + 4 - 4x^2 = -3x^2 -2x + 5.

You may use the derivative test to find local maximum of the function, but you may just remember from algebra that the x coordinate of the vertex of a quadratic function is at x= - b/(2a) = - 1/3

Substituting this to the equation of the ellipse you find y = +-4/3*sqr(3)

Therefore the 2 points of the ellipse furthest from the give point are: (-1/3, +-4/3*sqr(2))