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10-1.Circle and System of Circles
hard
The tangent at $P$, any point on the circle ${x^2} + {y^2} = 4$, meets the coordinate axes in $A$ and $B$, then
A
Length of $ AB$ is constant
B
$PA$ and $PB$ are always equal
C
The locus of the mid point of $AB$ is ${x^2} + {y^2} = {x^2}{y^2}$
D
None of these
Solution
(c) Let $P({x_1},\;{y_1})$ be a point on ${x^2} + {y^2} = 4$.
Then the equation of the tangent at $P$ is $x{x_1} + y{y_1} = 4$ which meets the coordinate axes at $A\left( {\frac{4}{{{x_1}}},\;0} \right)$ and $B{\rm{ }}\left( {0,\;\frac{4}{{{y_1}}}} \right)$.
Obviously, $(a)$ and $(b)$ are not true.
Let $(h,\;k)$ be the mid-point of $AB$.
Therefore $h = \frac{2}{{{x_1}}},\;k = \frac{2}{{{y_1}}}$
$i.e.$, ${x_1} = \frac{2}{h},\;{y_1} = \frac{2}{k}$
But $({x_1},\;{y_1})$ lies on ${x^2} + {y^2} = 4$.
Standard 11
Mathematics
