Question

find the equation of the ellipse whose centre is (-2,3) and whose semi-axis are 3 and 2 when major axis is (i) parallel to x-axis (ii) parallel to y-axis.

Answer 1

(i) When the major axis is parallel to the x-axis Let $\frac{(x-x_1{)}^2}{a^2}+\frac{(y-y_1{)}^2}{b^2}=1$ ...(1)
Here, $x_1$ and $y_1$ are -2 and 3, respectively, and 3 and 2 are the lengths of the axes.
Substituting the value in eq. (1), we get, $\frac{(x+2{)}^2}{9}+\frac{(y-3{)}^2}{4}=1$
$\Rightarrow \frac{4(x^2+4+4x)+9(y^2+9-6y)}{36}=1$
$\Rightarrow 4x^2+16+16x+9y^2+81-54y=36$
$\Rightarrow 4x^2+9y^2+16x-54y+61=0$
This is the required equation of the ellipse.
(ii) When the major axis is parallel to the y-axis
Let $\frac{(x-x_1{)}^2}{b^2}+\frac{(y-y_1{)}^2}{a^2}=1$ ...(1)
Here, $x_1$ and $y_1$ are -2 and 3, respectively, and 3 and 2 are the lengths of the axes.
Substituting the value in eq. (1), we get:
$\frac{(x+2{)}^2}{4}+\frac{(y-3{)}^2}{9}=1$
$\Rightarrow \frac{9(x^2+4+4x)+4(y^2+9-6y)}{36}=1$
$\Rightarrow 9x^2+36+36x+4y^2+36-24y=36$
$\Rightarrow 9x^2+4y^2+36x-24y+36=0$
This is the required equation of the ellipse.