Solving Problems Involving Different Conic Sections

Question

Direction: A problem is given with an answer, give the missing parts of the solution.

1. Find the standard equation of the parabola opening to the left whose axis contains the major axis of the ellipse 𝑥^2 + 4𝑦^2 − 10𝑥 − 24𝑦 + 45 = 0, whose focus is the center of the ellipse, and which passes through the covertices of this ellipse.

Solution: The standard form of the ellipse is (𝑥 − 5)^2/16 + (𝑦 − 3)^2/4 = 1

Its center (5,3) is the focus of the parabola.

Since 𝑏 = ______, its covertices are 𝑊1(___,___) and 𝑊2(___,___).

The vertex of the parabola, c units to the right of (5, 3), is (________,____).

Its equation can be written as (𝑦 − 3) 2 = ____________.

Since (5, 5) is a point on this parabola, we have (5 − 3) 2 = ____________

Solving this equation for 𝑐 > 0 yields 𝑐 = ______.

Therefore, the standard equation of the required parabola is (𝑦 − 3) 2 = 4(𝑥 − 6).

Answer 1

For the ellipse, b = 2, covertices are (5,1) and (5,5)

The focal length of the parabola (4p), which is the width at the focus is 4, therefore p = 1and the vertex is 1 unit to the right of the focus.

Vertex of parabola is (6,3)

p is actually -1 since parabola opens to the left, therefore 4p = 4(-1) = -4.

So we have (y-3)^2 = -4(x-6)