Question

If the lengths of semi-major and semi-minor axes of an ellipse are 2 and \[\sqrt{3}\] and their corresponding equations are y − 5 = 0 and x + 3 = 0, then write the equation of the ellipse. 

Answer 1

\[\text{ The lengths of the semi major and the semi minor axes of the ellipse are 2 and }\sqrt{3}, \text{ respectively }. \text{ Their corresponding equations are } y-5=0 \text{ and } x+3 =0.\]
\[So,a=2 \text{ and }b=\sqrt{3}\]
\[\text{ The equation of the ellipse is given by }\]
\[\frac{\left( x + 3 \right)^2}{4} + \frac{\left( y - 5 \right)^2}{3} = 1\]
\[ \Rightarrow 3 \left( x + 3 \right)^2 + 4 \left( y - 5 \right)^2 = 12\]
\[ \Rightarrow 3\left( x^2 + 9 + 6x \right) + 4\left( y^2 + 25 - 10y \right) = 12\]
\[ \Rightarrow 3 x^2 + 27 + 18x + 4 y^2 + 100 - 40y = 12\]
\[ \Rightarrow 3 x^2 + 4 y^2 + 18x - 40y + 115 = 0\]