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(C) Let the point on the line be $P(x_1, y_1)$. Since $P$ lies on the line $2x + y = 4$,we have $2x_1 + y_1 = 4$ ... $(1)$.The equation of the chord o

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$4 (x^2 + y^2) = 2x + y$

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(C) Let the point on the line be $P(x_1, y_1)$. Since $P$ lies on the line $2x + y = 4$,we have $2x_1 + y_1 = 4$ ... $(1)$.The equation of the chord of contact of tangents drawn from $P(x_1, y_1)$ to the unit circle $x^2 + y^2 = 1$ is $xx_1 + yy_1 = 1$.Let $(h, k)$ be the midpoint of this chord. The equation of a chord with midpoint $(h, k)$ for the circle $x^2 + y^2 = 1$ is $xh + yk = h^2 + k^2$.Comparing the two equations of the same chord:$\frac{x_1}{h} = \frac{y_1}{k} = \frac{1}{h^2 + k^2}$Thus,$x_1 = \frac{h}{h^2 + k^2}$ and $y_1 = \frac{k}{h^2 + k^2}$.Substituting these into equation $(1)$:$2 \left( \frac{h}{h^2 + k^2} \right) + \frac{k}{h^2 + k^2} = 4$$2h + k = 4(h^2 + k^2)$Replacing $(h, k)$ with $(x, y)$,the locus is $4(x^2 + y^2) = 2x + y$.

Tangents are drawn to a unit circle with centre at the origin from each point on the line $2x + y = 4$. Then the equation to the locus of the middle point of the chord of contact is

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