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10-2. Parabola, Ellipse, Hyperbola
hard
The angle between the pair of tangents drawn from the point $(1, 2)$ to the ellipse $3{x^2} + 2{y^2} = 5$ is
A
${\tan ^{ - 1}}(12/5)$
B
${\tan ^{ - 1}}(6/\sqrt 5 )$
C
${\tan ^{ - 1}}(12/\sqrt 5 )$
D
${\tan ^{ - 1}}(6/5)$
Solution
(c) The combined equation of the pair of tangents drawn from $(1,2)$ to the ellipse $3{x^2} + 2{y^2} = 5$ is
$(3{x^2} + 2{y^2} – 5)\,(3 + 8 – 5) = {(3x + 4y – 5)^2}$, [using $S'=T^2$]
$ \Rightarrow $ $9\,{x^2} – 24xy – 4{y^2} + ….. = 0$
The angle between the lines given by this equation is
$\tan \theta \, = \,\frac{{2\sqrt {{h^2} – ab} }}{{a + b}}$, where $a = 9,\,h = – 12,\,b = – 4$
$ \Rightarrow $$\tan \theta = 12\sqrt 5 \Rightarrow \,\theta = {\tan ^{ – 1}}(12/\sqrt 5 ).$
Standard 11
Mathematics
