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10-2. Parabola, Ellipse, Hyperbola
hard
જો પ્રમાણિત અતિવલયની ઉત્કેન્દ્ર્તા $2$ હોય જે બિંદુ $(4, 6)$ માંથી પસાર થતું હોય તો બિંદુ $(4, 6)$ આગળ અતિવલયનો સ્પર્શક મેળવો.
A
$2x -3y + 10 = 0$
B
$x -2y + 8 = 0$
C
$2x -y -2 = 0$
D
$3x -2y = 0$
(JEE MAIN-2019)
Solution
Let equation of hyperbola be $\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}} = 1\,\,$
passes through $\left( {4,6} \right)$
$ \Rightarrow \frac{{16}}{{{a^2}}} – \frac{{36}}{{{b^2}}} = 1\,\,\,\,\,\,\,…..\left( i \right)$
Also ${e^2} = 1 + \frac{{{b^2}}}{{{a^2}}} \Rightarrow {b^2} = 3{a^2}\,\,\,\,\,\,\,\,\,……\left( {ii} \right)$
from $(i)$ and $(ii)$
${a^2} = 4,{b^2} = 12$
equation $\frac{{{x^2}}}{4} – \frac{{{y^2}}}{{12}} = 1\,$
Tangent at $\left( {4,6} \right)$ is $x – \frac{y}{2} = 1\,\,\,$
Or
$2x – y = 2$
Standard 11
Mathematics
