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CONIC SECTIONS — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSCONIC SECTIONS4 Marks
Question
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse. $16x^2 + y^2 = 16$
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Answer

The given equation is $16x^2 + y^2 = 16$. It can be written as $16x^2 + y^2 = 16$ Or, $\frac{\text{x}^2}{1}+\frac{\text{y}^2}{16}=1$Or $\frac{\text{x}^2}{1^2}+\frac{\text{y}^2}{4^2}=1\ .....\text{(i)}$
Here, the denominator of $\frac{\text{y}^2}{4^2}$ is greater than the denominator of $\frac{\text{x}^2}{1^2}$. Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis. On comparing equation (i) with $\frac{\text{x}^2}{\text{b}^2}+\frac{\text{y}^2}{\text{a}^2}=1,$ we obtain b = 1 and a = 4. $\therefore \text{c}=\sqrt{\text{a}^2-\text{b}^2}=\sqrt{16-1}=\sqrt{15}$ Therefore, The coordinates of the foci are $(0,\pm\sqrt{15})$ The coordinates of the vertices are $(0, \pm4)$ Length of major axis = 2a = 8 Length of minor axis = 2b = 2 Eccentricity, $\text{e}=\frac{\text{c}}{\text{a}}=\frac{\sqrt{15}}{4}$ Length of latus rectum $=\frac{2\text{b}^2}{\text{a}}=\frac{2\times1}{4}=\frac{1}{2}.$

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