Question

find the equation of an ellipse whose vertex is (0,7) and directrix is the line y = 12

Answer 1

Answer:

As the vertex (0,7) lies on the y-axis, then the major axis is along the y-axis.

then, we have a=7 and the directrix is the line i.e, y=12 or $\frac{a}{e} =12$; where e is the eccentricity.

⇒$\frac{7}{e} =12$ or $e=\frac{7}{12}$.

Since, $b^2=a^2(1-e^2)$ we have,

$b^2=(7)^2(1-(\frac{7}{12})^2)$

⇒$b^2=49\cdot(1-\frac{49}{144})$

On Simplify:

$b^2=\frac{4655}{144}$

As, we know the equation of ellipse:

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

Substituting the values of $b^2=\frac{4655}{144}$ and  a=7 in above equation, we have:

$\frac{144x^2}{4655}+\frac{y^2}{7^2}=1$ or

$\frac{144x^2}{4655}+\frac{y^2}{49}=1$ or

$144x^2+95y^2=4655$

Therefore, the equation of an ellipse is, $144x^2+95y^2=4655$

Answer 2

Given:

Vertex = ( 0, 7 )

Directrix is the line y = 12

To find:

The equation of the ellipse.

Solution:

The general equation of the ellipse,

x^2 / a^2 + y^2 / b^2 = 1

As vertex is on the y axis,

a = 7

And,

The directrix,

a/e = 12

Where,

e is the eccintricity

Substituting and finding the value of e,

7/e = 12

Therefore,

e = 7/12

From the formula,

b^2 = a^2 ( 1 - e^2 )

b^2 = (7)^2 ( 1 - (7/12)^2)

b^2 = 4655/144

Substituting b^2 and a^2 in the equation of the ellipse,

144 x^2/4655 + y^2 / 49 = 1

We get,

144 x^2 + 95 y^2 = 4655

The equation of the ellipse, 144 x^2 + 95 y^2 = 4655