Let p be the number of all triangles that can be formed by joining the vertices of a regular polygon P on 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 ellip

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Solution

Q.
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: [2025]

1 $\frac{\sqrt{7}}{4}$

2 $\frac{1}{2}$

3 $\frac{1}{\sqrt{2}}$

4 $\frac{3}{4}$

Ans.
(3)

Total triangles $= {}^{n}C_{3}$

Total quadrilaterals $= {}^{n}C_{4}$

$∵ {}^{n}C_{3} + {}^{n}C_{4} = 126 \Rightarrow {}^{n + 1}C_{4} = 126$

$\Rightarrow n = 8$

The ellipse is $\frac{x^{2}}{16} + \frac{y^{2}}{n} = 1 \Rightarrow \frac{x^{2}}{16} + \frac{y^{2}}{8} = 1$

$∴ e = \sqrt{1 - \frac{8}{16}} = \sqrt{\frac{8}{16}} = \frac{1}{\sqrt{2}}$

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