Question

Find the standard form of the equation of the elipse and give the location of its fool The standard form of the equation of the ellipse is The foc are located at (Type ordered pairs. Use a comma to separate answers as needed. Simplify your answers Type an exacting radicals as needed) 2

Answer 1

Solution-

The given figure represents the vertical ellipse with

End points of major axis (0, ±a) = (0, ± 4) ,

End points of minor axis (±b, 0) = (±3, 0) and

Centre of the ellipse = (h, k) = (0,0).

We know the standard equation of vertical ellipse is

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

On putting the values (h, k) = (0,0) , a = 4 and b= 3, we get the equation of ellipse as

$\frac{(x-0)^2}{(3)^2}+\frac{(y-0)^2}{(4)^2}=1$

$\frac{x^2}{9}+\frac{y^2}{16}=1$

Now, we also know that

$c=\sqrt{a^2-b^2}$

So, on putting a =4 and b = 3, we get

$c=\sqrt{(4)^2-(3)^2}=\sqrt{16+9}=\sqrt{25}=5$

So,

So, foci of ellipse are located at (0, ±c) = (0, ±5).

Hence, the equation of ellipse is

$\frac{x^2}{9}+\frac{y^2}{16}=1$

and the location of foci are (0, -5), (0, 5) .