First slide

Definition of a circle

Question

The tangent at P, any point on the circle $x^{2} + y^{2} = 4,$ meets the coordinate axes in $A$ and $B,$ then

Moderate

A

length of $A B$ is constant
B

$P A$ and $P B$ are always equal
C

the locus of the mid-point of $A B$ is $x^{2} + y^{2} = x^{2} y^{2}$
D

none of these

Solution

Let $P (x_{1}, y_{1})$ be a point on $x^{2} + y^{2} = 4 .$ Then, the equation of the tangent at $P$ is $x x_{1} + y y_{1} = 4$
This meets the coordinate axes at $A (4 / x_{1}, 0)$ and $B (0, 4 / y_{1}) .$ Obviously $(a)$ and $(b)$ are not true. Let $(h, k)$ be the mid-point of $A B .$ Then,
$h = \frac{2}{x_{1}}, k = \frac{2}{y_{1}} \Rightarrow x_{1} = \frac{2}{h}, y_{1} = \frac{2}{k}$
Since, $(x_{1}, y_{1})$ lies on $x^{2} + y^{2} = 4$
$∴ \frac{4}{h^{2}} + \frac{4}{k^{2}} = 4 \Rightarrow \frac{1}{h^{2}} + \frac{1}{k^{2}} = 1$
Hence, the locus of $(h, k)$ is $\frac{1}{x^{2}} + \frac{1}{y^{2}} = 1,$ i.e. $x^{2} + y^{2} = x^{2} y^{2} .$

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