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10-1.Circle and System of Circles
hard
The absolute difference between the squares of the radii of the two circles passing through the point $(-9,4)$ and touching the lines $x+y=3$ and $x-y=3$, is equal to ______.
A$768$
B$254$
C$654$
D$147$
(JEE MAIN-2025)
Solution
Centre $(a, 0)$$r=\left|\frac{a-0-3}{\sqrt{2}}\right|$
$\operatorname{circle}(x-a)^2+y^2=\left(\frac{a-3}{\sqrt{2}}\right)^2$
passes through $(-9,4)$
$2\left(a^2+18 a+81+16\right)=\left(a^2-6 a+9\right)$
$a^2+42 a+185=0$
$(a+37)(a+5)=0$
$\Rightarrow a=-37,-5$
$r_1=\left|\frac{-37-3}{\sqrt{2}}\right|=20 \sqrt{2}$
$r_2=\left|\frac{-5-3}{\sqrt{2}}\right|=4 \sqrt{2}$
$\left| r _1^2- r _2^2\right|=|800-32|=768$
Standard 11
Mathematics
