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JEE MainMathematicsModerateCircles

Equation of circle touching the line x + y = 4 at (1,3) and
intersecting the circle x2+y2=4x^{2} + y^{2} = 4 orthogonally, is

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Circle is (x−1)2+(y−3)2+λ(x+y−4)=0(x - 1)^{2} + (y - 3)^{2} + \lambda(x + y - 4) = 0
⇒x2+y2+x(λ+2)+y(λ−6)+(10−4λ)=0\Rightarrow x^{2} + y^{2} + x(\lambda + 2) + y(\lambda - 6) + (10 - 4\lambda) = 0
2g1g2+2f1f2=4+λ2⇒0+0=10−4λ−42g_{1}g_{2} + 2f_{1}f_{2} = 4 + \lambda_{2} \Rightarrow 0 + 0 = 10 - 4\lambda - 4
⇒λ=32⇒x2+y2−x2−9y2+4=0\Rightarrow \lambda = \frac{3}{2} \Rightarrow x^{2} + y^{2} - \frac{x}{2} - \frac{9y}{2} + 4 = 0

Source: Grass-G / M-25 Mathematics Paper 7 Jan.docx

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