A pair of tangents is drawn from every point | vedclass

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(D) Let $P(x_1, y_1)$ be a point on the line $3x + 4y = 12$. The equation of the chord of contact of the point $P(x_1, y_1)$ with respect to the circl

Vedclass · Accepted Answer

$\left(1, \frac{4}{3}\right)$

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(D) Let $P(x_1, y_1)$ be a point on the line $3x + 4y = 12$. The equation of the chord of contact of the point $P(x_1, y_1)$ with respect to the circle $x^2 + y^2 = 4$ is given by $x x_1 + y y_1 = 4$,which can be written as $x x_1 + y y_1 - 4 = 0$. Since $P(x_1, y_1)$ lies on the line $3x_1 + 4y_1 = 12$,we can rewrite this as $3x_1 + 4y_1 - 12 = 0$,or dividing by $3$,$x_1 + \frac{4}{3} y_1 - 4 = 0$. Comparing the equation of the chord of contact $x x_1 + y y_1 - 4 = 0$ with the line $x_1 + \frac{4}{3} y_1 - 4 = 0$,we observe that the chord of contact passes through the fixed point $(x, y) = (1, \frac{4}{3})$. Thus,the variable chord of contact always passes through the fixed point $\left(1, \frac{4}{3}\right)$.

$A$ pair of tangents is drawn from every point on the line $3x + 4y = 12$ to the circle $x^2 + y^2 = 4$. Their variable chord of contact always passes through a fixed point whose coordinates are:

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