The locus of the midpoint of the chord of con | vedclass

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Question

(A) Let the midpoint of the chord of contact be $(h, k)$.The equation of the chord of contact with midpoint $(h, k)$ for the circle $x^2 + y^2 = 9$ is

Vedclass · Accepted Answer

$20(x^2 + y^2) - 36x + 45y = 0$

Vedclass · Answer

(A) Let the midpoint of the chord of contact be $(h, k)$.The equation of the chord of contact with midpoint $(h, k)$ for the circle $x^2 + y^2 = 9$ is given by $hx + ky = h^2 + k^2$ ... $(1)$.Let a point on the line $4x - 5y = 20$ be $P(\alpha, \beta)$,where $4\alpha - 5\beta = 20$,so $\beta = \frac{4\alpha - 20}{5}$.The equation of the chord of contact from point $P(\alpha, \beta)$ to the circle $x^2 + y^2 = 9$ is $\alpha x + \beta y = 9$ ... $(2)$.Comparing $(1)$ and $(2)$,we have $\frac{h}{\alpha} = \frac{k}{\beta} = \frac{h^2 + k^2}{9}$.Thus,$\alpha = \frac{9h}{h^2 + k^2}$ and $\beta = \frac{9k}{h^2 + k^2}$.Substitute these into the line equation $4\alpha - 5\beta = 20$:$4\left(\frac{9h}{h^2 + k^2}\right) - 5\left(\frac{9k}{h^2 + k^2}\right) = 20$.$36h - 45k = 20(h^2 + k^2)$.Replacing $(h, k)$ with $(x, y)$,the locus is $20(x^2 + y^2) - 36x + 45y = 0$.

The locus of the midpoint of the chord of contact of tangents drawn from a point on the line $4x - 5y = 20$ to the circle $x^2 + y^2 = 9$ is:

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