Let p be the number of all triangles that can be formed by joinin… - Vidyadip

MCQ

Let $p$ be the number of all triangles that can be formed by joining the vertices of a regular polygon P of n sides and q be the number of all quadrilaterals that can be formed by joining the vertices of $P$. If $p+q=126$, then the eccentricity of the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{n}=1$ is :

A
$\frac{3}{4}$
B
$\frac{1}{2}$
C
$\frac{\sqrt{7}}{4}$
D
$\frac{1}{\sqrt{2}}$

✓

Answer

D. $\frac{1}{\sqrt{2}}$
Total trangles $=\Rightarrow={ }^{h} \mathrm{C}_{3}$
Total auadrilaterals $={ }^{h} \mathrm{C}_{4}=\mathrm{q}$
${ }^{\mathrm{n}} \mathrm{C}_{3}+{ }^{\mathrm{n}} \mathrm{C}_{4}=126 \Rightarrow{ }^{\mathrm{n}+1} \mathrm{C}_{4}=126$
$\Rightarrow \mathrm{n}+1=9 \Rightarrow \mathrm{n}=8$
$\frac{x^{2}}{16}+\frac{y^{2}}{n}=1 \Rightarrow \frac{x^{2}}{16}+\frac{y^{2}}{8}=1$
$\mathrm{e}=\sqrt{1-\frac{8}{16}}=\sqrt{\frac{8}{16}}=\frac{1}{\sqrt{2}}$

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