The locus of the mid-point of the chord of co | vedclass

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(A) Let $P(t, \frac{4t - 20}{5})$ be a point on the line $4x - 5y = 20$.The equation of the chord of contact of tangents drawn from $P(t, \frac{4t - 2

Vedclass · Accepted Answer

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

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(A) Let $P(t, \frac{4t - 20}{5})$ be a point on the line $4x - 5y = 20$.The equation of the chord of contact of tangents drawn from $P(t, \frac{4t - 20}{5})$ to the circle $x^2 + y^2 = 9$ is given by $T = 0$:$tx + (\frac{4t - 20}{5})y = 9$  --- $(1)$Let $M(h, k)$ be the mid-point of this chord. The equation of a chord of a circle with mid-point $(h, k)$ is given by $T = S_1$:$hx + ky = h^2 + k^2$  --- $(2)$Comparing equations $(1)$ and $(2)$,we have:$\frac{t}{h} = \frac{4t - 20}{5k} = \frac{9}{h^2 + k^2}$From $\frac{t}{h} = \frac{9}{h^2 + k^2}$,we get $t = \frac{9h}{h^2 + k^2}$.From $\frac{4t - 20}{5k} = \frac{9}{h^2 + k^2}$,we get $4t - 20 = \frac{45k}{h^2 + k^2}$.Substituting $t$ in the second equation:$4(\frac{9h}{h^2 + k^2}) - 20 = \frac{45k}{h^2 + k^2}$$36h - 20(h^2 + k^2) = 45k$$20(h^2 + k^2) - 36h + 45k = 0$Replacing $(h, k)$ with $(x, y)$,the locus is:$20(x^2 + y^2) - 36x + 45y = 0$

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4x - 5y = 20$ to the circle $x^2 + y^2 = 9$ is

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