Question

Write an equation in standard form for each ellipse with center (0, 0) and co-vertex at (5, 0); focus at (0, 3).

Answer 1

Answer:

The required standard form of  ellipse is $\frac{x^2}{25}+\frac{y^2}{34}=1$.

Step-by-step explanation:

The standard form of an ellipse is

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

Where, (h,k) is center of the ellipse.

It is given that the center of the circle is (0,0), so the standard form of the ellipse is

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$             .... (1)

If a>b, then coordinates of vertices are (±a,0), coordinates of co-vertices are (0,±b) and focus (±c,0).

$c^2=a^2-b^2$            .... (2)

If a<b, then coordinates of vertices are (0,±b), coordinates of co-vertices are (±a,0) and focus (0,±c).

$c^2=b^2-a^2$            .... (3)

It is given that co-vertex of the ellipse at (5, 0); focus at (0, 3). So, a<b we get

$a=5,c=3$

Substitute a=5 and c=3 these values in equation (3).

$3^2=b^2-(5)^2$

$9=b^2-25$

$34=b^2$

$\sqrt{34}=b$

Substitute a=5 and $b=\sqrt{34}$ in equation (1), to find the required equation.

$\frac{x^2}{5^2}+\frac{y^2}{(\sqrt{34})^2}=1$

$\frac{x^2}{25}+\frac{y^2}{34}=1$

Therefore the required standard form of  ellipse is $\frac{x^2}{25}+\frac{y^2}{34}=1$.