Question

If the vertices of an ellipse are at (1, 5) and (1, -5) and (3, 0) is a point on the ellipse, what is the ellipse equation?

Answer 1

Answer:

$\dfrac{(x-1)^2}{4}+\dfrac{y^2}{25}=1$

Step-by-step explanation:

The equation of the ellipse is

$\dfrac{(x-x_0)^2}{a^2}+\dfrac{(y-y_0)^2}{b^2}=1,$

where $(x_0,y_0)$ are the coordinates of the center.

If the vertices of an ellipse are at A(1, 5) and B(1, -5), then the center is the midpoint of the segment AB. Hence, the center has coordinates

$\left(\dfrac{1+1}{2},\dfrac{5+(-5)}{2}\right)=(1,0).$

The coordinates of the vertices satisfy the equation:

$\dfrac{(1-1)^2}{a^2}+\dfrac{(5-0)^2}{b^2}=1\Rightarrow b^2=25.$

If (3, 0) is a point on the ellipse, then its coordinates satisfy the equation:

$\dfrac{(3-1)^2}{a^2}+\dfrac{(0-0)^2}{b^2}=1\Rightarrow a^2=4.$

Therefore, the equation of the ellipse is

$\dfrac{(x-1)^2}{4}+\dfrac{y^2}{25}=1.$

Answer 2

Answer:

the answer is b.

Step-by-step explanation: