# How do you find equation of ellipse with two vertices V1(7,12) and V2(7, -8), and passing through the point P(1,8)?

May 30, 2017

$\frac{4 {\left(x - 7\right)}^{2}}{225} + {\left(y - 2\right)}^{2} / 100 = 1$

#### Explanation:

The equation of an ellipse with centre $\left(h , k\right)$ and axis parallel to $x$-axis $2 a$ and axis parallel to $y$-axis $2 b$ is of the form ${\left(x - h\right)}^{2} / {a}^{2} + {\left(y - k\right)}^{2} / {b}^{2} = 1$.

Here two vertices are $\left(7 , 12\right)$ and $\left(7 , - 8\right)$, hence centre is $\left(7 , \frac{12 - 8}{2}\right)$ or $\left(7 , 2\right)$ and as distance between vertices (with common abscissa) is $12 - \left(- 8\right) = 20$ and hence $2 b = 20$ or $b = 10$.

Hence equation is ${\left(x - 7\right)}^{2} / {a}^{2} + {\left(y - 2\right)}^{2} / 100 = 1$

As it passes through $P \left(1 , 8\right)$

${\left(1 - 7\right)}^{2} / {a}^{2} + {\left(8 - 2\right)}^{2} / 100 = 1$

or $\frac{36}{a} ^ 2 + \frac{36}{100} = 1$

or $\frac{36}{a} ^ 2 = 1 - \frac{36}{100} = \frac{64}{100}$

Hence ${a}^{2} = \frac{36 \times 100}{64}$

and $a = \frac{60}{8} = 7.5$ and equation of ellipse is

$\frac{4 {\left(x - 7\right)}^{2}}{225} + {\left(y - 2\right)}^{2} / 100 = 1$

Follwing is the graph with three given points marked.

graph{((4(x-7)^2)/225+(y-2)^2/100-1)((x-1)^2+(y-8)^2-0.04)((x-7)^2+(y-12)^2-0.04)((x-7)^2+(y+8)^2-0.04)=0 [-15, 35, -10, 15]}
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