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10-1.Circle and System of Circles
medium
Line $y = x + a\sqrt 2 $ is a tangent to the circle ${x^2} + {y^2} = {a^2}$ at
A
$\left( {\frac{a}{{\sqrt 2 }},\frac{a}{{\sqrt 2 }}} \right)$
B
$\left( { - \frac{a}{{\sqrt 2 }}, - \frac{a}{{\sqrt 2 }}} \right)$
C
$\left( {\frac{a}{{\sqrt 2 }}, - \frac{a}{{\sqrt 2 }}} \right)$
D
$\left( { - \frac{a}{{\sqrt 2 }},\frac{a}{{\sqrt 2 }}} \right)$
Solution
(d) Suppose that the point be $(h, k)$.
Tangent at $(h, k)$ is $hx + ky = {a^2} \equiv x – y = – \sqrt 2 a$
or $\frac{h}{1} = \frac{k}{{ – 1}} = \frac{{{a^2}}}{{ – \sqrt 2 a}}$
or $h = – \frac{a}{{\sqrt 2 }},\;k = \frac{a}{{\sqrt 2 }}$
Therefore, point of contact is $\left( { – \frac{a}{{\sqrt 2 }},\;\frac{a}{{\sqrt 2 }}} \right)$.
Standard 11
Mathematics
