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10-1.Circle and System of Circles
hard
If a circle of radius $R$ passes through the origin $O$ and intersects the coordinate axes at $A$ and $B,$ then the locus of the foot of perpendicular from $O$ on $AB$ is
A
${({x^2} + {y^2})^2} = 4{R^2}{x^2}{y^2}$
B
${({x^2} + {y^2})^3} = 4{R^2}{x^2}{y^2}$
C
${({x^2} + {y^2})^2} = 4R{x^2}{y^2}$
D
$({x^2} + {y^2})(x + y) = {R^2}xy$
(JEE MAIN-2019)
Solution
Slope of $AB = \frac{{ – h}}{k}$
Equation of $AB$ is $hx + ky = {h^2} + {k^2}$
$A\left( {\frac{{{h^2} + {k^2}}}{h},0} \right),B\left( {0,\frac{{{h^2} + {k^2}}}{k}} \right)$
$As,AB = 2R$
$ \Rightarrow {\left( {{h^2} + {k^2}} \right)^3} = 4{R^2}{h^2}{k^2}$
$ \Rightarrow {\left( {{x^2} + {y^2}} \right)^3} = 4{R^2}{x^2}{y^2}$
Standard 11
Mathematics
