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10.2 parabola,ellipse,hyperbola — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS10.2 parabola,ellipse,hyperbola1 Mark
MCQ
$P(6, 3)$ is a point on the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ . If the normal at point $P$ intersect the $x-$ axis at $(10, 0)$ , then the eccentricity of the hyperbola is
  • ✓
    $\sqrt {\frac{5}{3}} $
  • B
    $\frac{{\sqrt {13} }}{3}$
  • C
    $\sqrt {\frac{5}{2}} $
  • D
    $\frac{{\sqrt {13} }}{2}$
✓

Answer

Correct option: A.
$\sqrt {\frac{5}{3}} $
a
$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$

Slope of normal is $-\frac{a^{2}}{2 b^{2}}$

$\therefore$ Equation of normal at $\mathrm{P}(6,3)$ is

$y-3=-\frac{a^{2}}{2 b^{2}}(x-6)$

at $(10,0) \Rightarrow-3=-\frac{a^{2}}{2 b^{2}}(10-6)$

$\Rightarrow \quad \frac{3}{2}=\frac{a^{2}}{b^{2}}$

$\therefore \quad e^{2}=1+\frac{b^{2}}{a^{2}}=1+\frac{2}{3}=\frac{5}{3}$

$\therefore \quad e=\sqrt{\frac{5}{3}}$

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