Question

The straight line $x - 2y + 1 = 0$ intersects the circle $x^2 + y^2 = 25$ in points T and T', find the co-ordinates of a point of intersection of tangents drawn at T and T' to the circle.

• A. (-25, 50)
• B. (25, -50)
• C. (25, 50)
• D. (-25, -50)

Answer 1

Answer: (A)

(-25, 50)

Answer 2

Let the point of intersection of the tangents be $(h, k)$. The equation of the chord of contact of tangents drawn from an external point $(h, k)$ to the circle $x^2 + y^2 = r^2$ is $hx + ky = r^2$. For the given circle $x^2 + y^2 = 25$, the equation of the chord of contact is $hx + ky = 25$. The given line $x - 2y + 1 = 0$ is the chord of contact. Rewriting the given line as $x - 2y = -1$. Comparing the coefficients of $hx + ky = 25$ and $x - 2y = -1$, we get: $\frac{h}{1} = \frac{k}{-2} = \frac{25}{-1}$ From this proportion, we find $h = -25$ and $k = 50$. Therefore, the coordinates of the point of intersection of the tangents are $(-25, 50)$.