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10-1.Circle and System of Circles
hard
ધારો કે રેખાઓ $y+2 x=\sqrt{11}+7 \sqrt{7}$ અને $2 y + x =2 \sqrt{11}+6 \sqrt{7}$ એ વર્તુળ $C:(x-h)^{2}+(y-k)^{2}=r^{2}$. ના અભિલંબ છે જો રેખા $\sqrt{11} y -3 x =\frac{5 \sqrt{77}}{3}+11$ એ વર્તુળ $C$, નો સ્પર્શક હોય તો $(5 h-8 k)^{2}+5 r^{2}$ નું મૂલ્ય ...................છે
A
$916$
B
$816$
C
$856$
D
$86$
(JEE MAIN-2022)
Solution
Normal are
$y +2 x =\sqrt{11}+7 \sqrt{7}$
$2 y + x =2 \sqrt{11}+6 \sqrt{7}$
Center of the circle is point of intersection of ormals i.e.
$\left(\frac{8 \sqrt{7}}{3}, \sqrt{11}+\frac{5 \sqrt{7}}{3}\right)$
Tangent is $\sqrt{11} y-3 x=\frac{5 \sqrt{77}}{3}+11$
Radius will be $\perp$ distance of tangent from center
i.e. $4 \sqrt{\frac{7}{5}}$
Now $(5 h -8 k )^{2}+5 r ^{2}=816$
Standard 11
Mathematics
