# Find center, focii and intercepts on x-axis and y-axis of ellipse 16x^2+9y^2+64x-18y-71=0?

Jul 7, 2017

Center is $\left(- 2 , 1\right)$, focii are $\left(- 2 , 1 - \sqrt{7}\right)$ and $\left(- 2 , 1 + \sqrt{7}\right)$. Intercepts on $x$-axis are $1 \pm \frac{4 \sqrt{5}}{3}$ and intercepts on $y$-axis are $- 2 \pm \frac{3 \sqrt{15}}{4}$.

#### Explanation:

The general form of equation of ellipse is

${\left(x - h\right)}^{2} / {a}^{2} + {\left(y - k\right)}^{2} / {b}^{2} = 1$, whose center is $\left(h , k\right)$ and major axis is along $x$-axis if $a > b$ and along $y$-axis if $a < b$.

Further eccentricity $e = \sqrt{1 - {b}^{2} / {a}^{2}}$, if $a > b$ or $e = \sqrt{1 - {a}^{2} / {b}^{2}}$, if $a < b$.

Focii are at major axis at a distance of $\pm a e$ or $= \pm b e$, again depending on $a > b$ or $b > a$.

We can express $16 {x}^{2} + 9 {y}^{2} + 64 x - 18 y - 71 = 0$ as

$16 {x}^{2} + 64 x + 9 {y}^{2} - 18 y - 71 = 0$

or $16 \left({x}^{2} + 4 x + 4\right) + 9 \left({y}^{2} - 2 y + 1\right) - 64 - 9 - 71 = 0$

or $16 {\left(x + 2\right)}^{2} + 9 {\left(y - 1\right)}^{2} = 144$

or ${\left(x + 2\right)}^{2} / {3}^{2} + {\left(y - 1\right)}^{2} / {4}^{2} = 1$

Hence center is $\left(- 2 , 1\right)$ and major axis along $y$-axis is $8$ and minor axis along $x$-axis is $6$.

Eccentricity is $e = \sqrt{1 - {\left(\frac{3}{4}\right)}^{2}} = \frac{\sqrt{7}}{4}$

and as $b e = \sqrt{7}$, focii are $\left(- 2 , 1 \pm \sqrt{7}\right)$

Intercepts on $x$-axis can be found by putting $y = 0$ i.e. $9 {y}^{2} - 18 y - 71 = 0$ $y = \frac{18 \pm \sqrt{324 - 4 \cdot 9 \cdot \left(- 71\right)}}{18} = 1 \pm \frac{\sqrt{324 + 2556}}{18} = 1 \pm 24 \frac{\sqrt{5}}{18} = 1 \pm \frac{4 \sqrt{5}}{3}$

Intercepts on $y$-axis can be found by putting $x = 0$ i.e. $16 {x}^{2} + 64 x - 71 = 0$ $x = \frac{- 64 \pm \sqrt{{64}^{2} - 4 \cdot 16 \cdot \left(- 71\right)}}{32} = - 2 \pm \frac{\sqrt{4096 + 4544}}{32} = - 2 \pm 24 \frac{\sqrt{15}}{32} = - 2 \pm \frac{3 \sqrt{15}}{4}$

graph{(16x^2+9y^2+64x-18y-71)((x+2)^2+(y-1-sqrt7)^2-0.03)((x+2)^2+(y-1+sqrt7)^2-0.03)=0 [-12.21, 7.79, -4.36, 5.64]}