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10-2. Parabola, Ellipse, Hyperbola
hard
बिंदु $(1,3)$ से दीर्घवृत्त $2 x^2+3 y^2=5$ पर डाली गई दो स्पर्श रेखाओं के बीच न्यून कोण है :
A
$\tan ^{-1}\left(\frac{16}{7 \sqrt{5}}\right)$
B
$\tan ^{-1}\left(\frac{24}{7 \sqrt{5}}\right)$
C
$\tan ^{-1}\left(\frac{32}{7 \sqrt{5}}\right)$
D
$\tan ^{-1}\left(\frac{3+8 \sqrt{5}}{35}\right)$
(JEE MAIN-2022)
Solution
Equation of tangent to the ellipse $2 x ^{2}+3 y ^{2}=5$ is
$y = mx \pm \sqrt{\frac{5}{2} m ^{2}+\frac{5}{3}}$
It pass through $(1,3)$
$3=m \pm \sqrt{\frac{5}{2} m^{2}+\frac{5}{3}}$
$3\,m^{2}+12\,m-\frac{44}{3}=0$
Let $\theta$ be the angle between the tangents
$\tan \theta=\left|\frac{m_{1}-m_{2}}{1+m_{1} m_{2}}\right|$
$\tan \theta=\left|\frac{3 \sqrt{320}}{-35}\right|$
$\theta=\tan ^{-1}\left(\frac{24}{7 \sqrt{5}}\right)$
Standard 11
Mathematics
