Gujarat BoardEnglish MediumSTD 11 ScienceMATHSEllipse4 Marks
Question
Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:$\text{x}^2+4\text{y}^2-2\text{x}=0$
✓
Answer
We have,$\text{x}^2+4\text{y}^2-2\text{x}=0$
$\Rightarrow\text{x}^2-2\text{x}+4\text{y}^2=0$
$\Rightarrow\Big(\text{x}^2-2\text{x}+1^2-1^2\Big)+4\text{y}^2=0$
$\Rightarrow\big(\text{x}-1\big)^2-1+4\text{y}^2=0$
$\Rightarrow\big(\text{x}-1\big)^2+4\text{y}^2=1$
$\Rightarrow\frac{\big(\text{x}-1\big)^2}{1}+\frac{\text{y}^2}{\frac{1}{4}}=1$
$\Rightarrow\frac{\big(\text{x}-1\big)^2}{1^2}+\frac{\text{y}^2}{\Big(\frac{1}{2}\Big)^2}=1\ \dots(\text{i})$
$\therefore\ $The coordinates of centre of the ellipse are (1, 0).
Shifting the origin at (1, 0) without rotating the coordinate axes and denoting the new coordinates with respect to the new axes by x and y, we havex = x + 1 and y = y $\dots(\text{ii})$
using these relations, equation (i) reduces to
$\Rightarrow\frac{\text{x}^2}{1^2}+\frac{\text{y}^2}{\Big(\frac{1}{2}\Big)^2}=1,$ where $\text{a}=1$ and $\text{b}=\frac{1}{2}$
clearly, a > b. so, the given equation represents an ellipse whose and minor axes are along x and y axes respectively. Length of the axes:Major-axis = $2\times\text{a}=2\times1=2$
and, Minor-axis = $2\times\text{b}=2\times\frac{1}{2}=1$
Eccentricity: The eccentricity e is given by
$\text{e}=\sqrt{1-\frac{\text{b}^2}{\text{a}^2}}$
$=\sqrt{1-\frac{1}{\frac{4}{1}}}$
$=\sqrt{1-\frac{1}{{4}}}$
$=\sqrt{\frac{3}{4}}$
$=\frac{\sqrt{3}}{2}$
foci: The coordinates of the foci with respect to the new axes are given by $\big(\text{x}=\pm\text{ae},\ \text{y}=0\big)$ i.e., $\big(\text{x}=\pm\frac{\sqrt{3}}{2},\ \text{y}=0\big)$
Putting $\text{x}=\pm\frac{\sqrt{3}}{2}$ and $\text{y}=0$ in equation (iii), we get
$\text{x}=\pm\frac{\sqrt{3}}{2}+1$ and $\text{y}=0$
$\text{x}=1\pm\frac{\sqrt{3}}{2}$ and $\text{y}=0$
So, the coordinates of foci with respect to the old axes given by $\Big(1\pm\frac{\sqrt{3}}{2},\ 0\Big).$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.