Gujarat BoardEnglish MediumSTD 11 ScienceMATHSEllipse3 Marks
Question
Find the equation to the ellipse whose centre is (-2, 3) and semi-axis are 3 and 2 when major axis is Parallel to y-axis.
✓
Answer
Let 2a and 2b the major and minor axes of the ellipse. Then, its equation is$\frac{(\text{x}+2)^2}{\text{a}^2}+\frac{(\text{y}-3)^2}{\text{b}^2}=1$ $\big[\because\ $centre: (-2, 3) $\dots(\text{i})\ \big]$
we have, semi-major axis = a = 2$\Rightarrow\text{a}^2=4$
and semi-major axis = b = 3$\Rightarrow\text{b}^2=4$
Putting $a^2 = 4$ and $b^2 = 9$ in equation (i), we get$\frac{(\text{x}+2)^2}{4}+\frac{(\text{y}-3)^2}{9}=1$
$\Rightarrow\frac{9(\text{x}+2)^2+4(\text{y}-3)^2}{36}=1$
$\Rightarrow9(\text{x}+2)^2+4(\text{y}-3)^2=36$
$\Rightarrow9\big[\text{x}^2+4+4\text{x}\big]+4\big[\text{y}^2+9-6\text{y}\big]=36$
$\Rightarrow9\text{x}^2+36+36\text{x}+4\text{y}^2+36-24\text{y}=36$
$\Rightarrow9\text{x}^2+4\text{y}^2+36\text{x}-24\text{y}+36+36-36=0$
$\Rightarrow9\text{x}^2+4\text{y}^2+36\text{x}-24\text{y}+36=0$
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