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10-1.Circle and System of Circles
hard
A pair of tangents are drawn from the origin to the circle ${x^2} + {y^2} + 20(x + y) + 20 = 0$. The equation of the pair of tangents is
A
${x^2} + {y^2} + 10xy = 0$
B
${x^2} + {y^2} + 5xy = 0$
C
$2{x^2} + 2{y^2} + 5xy = 0$
D
$2{x^2} + 2{y^2} - 5xy = 0$
Solution
(c) Equation of pair of tangents is given by $S{S_1} = {T^2}$.
Here $S = {x^2} + {y^2} + 20{\rm{ }}(x + y) + 20,\;\;{S_1} = 20$
$T = 10(x + y) + 20$
$\therefore \;S{S_1} = {T^2}$
$ \Rightarrow 20\,\{ {x^2} + {y^2} + 20(x + y) + 20\} = {10^2}{(x + y + 2)^2}$
$ \Rightarrow 4{x^2} + 4{y^2} + 10xy = 0 \Rightarrow 2{x^2} + 2{y^2} + 5xy = 0$.
Standard 11
Mathematics
