The tangent at P,any point on the circle {x^2 | vedclass

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(C) Let $P(x_1, y_1)$ be a point on the circle $x^2 + y^2 = 4$. The equation of the tangent at $P$ is $xx_1 + yy_1 = 4$. This tangent meets the coordi

Vedclass · Accepted Answer

The locus of the mid-point of $AB$ is $\frac{1}{x^2} + \frac{1}{y^2} = \frac{1}{4}$

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(C) Let $P(x_1, y_1)$ be a point on the circle $x^2 + y^2 = 4$. The equation of the tangent at $P$ is $xx_1 + yy_1 = 4$. This tangent meets the coordinate axes at $A(\frac{4}{x_1}, 0)$ and $B(0, \frac{4}{y_1})$. Let $(h, k)$ be the mid-point of $AB$. Then $h = \frac{2}{x_1}$ and $k = \frac{2}{y_1}$,which implies $x_1 = \frac{2}{h}$ and $y_1 = \frac{2}{k}$. Since $(x_1, y_1)$ lies on $x^2 + y^2 = 4$,we have $(\frac{2}{h})^2 + (\frac{2}{k})^2 = 4$. Dividing by $4$,we get $\frac{1}{h^2} + \frac{1}{k^2} = 1$. Replacing $(h, k)$ with $(x, y)$,the locus is $\frac{1}{x^2} + \frac{1}{y^2} = 1$.

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

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