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10-1.Circle and System of Circles
hard
ધારો કે $y=x+2,4 y=3 x+6^2 y^2 3 y=4 x+1$ અને $3 y=4 x+1$ એ વર્તુળ $(x- h )^2+(y- k )^2= r ^2$ ની ત્રણ સ્પર્શ રેખાઓ છે.તો $h+k=..........$
A
$5$
B
$5(1+\sqrt{2})$
C
$6$
D
$5 \sqrt{2}$
(JEE MAIN-2023)
Solution
$L _1: y=x+2, L_2: 4 y=3 x+6, L_3: 3 y=4 x+1$
Bisector of lines $L _2 \& L _3$
$\frac{4 x-3 y+1}{5}=\pm\left(\frac{3 x-4 y+6}{5}\right)$
$\text { (+) } 4 x-3 y+1=3 x-4 y+6$
$x+y=5$
Centre lies on Bisector of $4 x-3 y+1=0 \&$
$\text { (0) } 3 x-4 y+6=0$
$\Rightarrow h + k =5$
Standard 11
Mathematics
