Question

The equation of the ellipse whose centre is at originand which passes through the points $(-3,1)$ and $(2,-2)$ is $(a>b)-$

• A. $5x^2+3y^2=32$
• B. $3x^2+5y^2=32$
• C. $5x^2-3y^2=32$
• D. $3x^2+5y^2+32=0$

Answer 1

The correct option is B $3x^2+5y^2=32$

$\begin{array}{c}Theequationofellipsehavingcentre\\ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\\ for\left(-3,1\right)\Rightarrow \frac{9}{a^2}+\frac{1}{b^2}=1\\ for\left(2,-2\right)\Rightarrow \frac{4}{a^2}+\frac{4}{b^2}=1\\ putx=\frac{1}{a^2}\\ y=\frac{1}{b^2}\\ 9x+y=1\to \left(i\right)\\ 4x+4y=1\to \left(ii\right)\\ multiplyequalby4andsubstractequation\left(ii\right)from\left(i\right)\\ weget,\\ 36x+4y=4\\ 4x+4y=1\\ ---\\ \overline{32x=3}\\ x=\frac{3}{32}\\ \therefore y=1-9x=1-\frac{27}{32}=\frac{5}{32}\\ x=\frac{1}{a^2}=\frac{3}{32}\\ y=\frac{1}{b^2}=\frac{5}{32}\\ \frac{x^2}{a^2}+\frac{y^2}{b^2}=2\\ \frac{3}{32}{x}^2+\frac{5}{32}{y}^2=1\end{array}$