Question

: Which of the following equations represents an ellipse having vertices located at (2,9) and (2,-5) and foci located at (2,5) and (2,-1)

A: (x+2)^2 (y+2)^2
---------- + ---------- = 1
40 49

B: (x-2)^2 (y-2)^2
--------- + ---------- = 1
9 49

C: (x-2)^2 (y-2)^2
--------- + ---------- = 1
40 49

D: (x-2)^2 (y-2)^2
---------- + ----------- = 1
49 40

Answer 1

Answer on Question #57276 – Math – Analytic Geometry

Question

Which of the following equations represents an ellipse having vertices located at (2,9) and (2,-5) and foci located at (2,5) and (2,-1)?

A: $(x+2)^2 (y+2)^2$

--- + --- = 1

40 49

B: $(x-2)^2 (y-2)^2$

--- + --- = 1

9 49

C: $(x-2)^2 (y-2)^2$

--- + --- = 1

40 49

D: $(x-2)^2 (y-2)^2$

--- + --- = 1

49 40

Solution

For ellipse $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$, $a > b$

Vertices: $(h + a, k), (h - a, k)$,

Foci: $(h + c, k), (h - c, k)$, where $c^2 = a^2 - b^2$.

For ellipse $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$ , $a < b$

Vertices: $(h,k + b),(h,k - b)$

Foci: $(h,k + c),(h,k - c)$ , where $c^2 = b^2 -a^2$

If it is given that vertices are located at (2,9) and (2,-5) and foci are located at (2,5) and (2,-1), then the second type of ellipses $(a < b)$ meets these conditions

$$h = 2, \qquad \left\{ \begin{array}{l} k + b = 9 \\ k - b = - 5 ^ {\prime} \end{array} \right. \left\{ \begin{array}{l} k + c = 5 \\ k - c = - 1 \end{array} \right. \to k = 2, b = 7, c = 3,$$ $$a ^ {2} = b ^ {2} - c ^ {2} = 4 9 - 9 = 4 0.$$

Equation of ellipse is $\frac{(x - 2)^2}{40} + \frac{(y - 2)^2}{49} = 1$ .

Answer: C. $\frac{(x - 2)^2}{40} + \frac{(y - 2)^2}{49} = 1$ .

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