Middle point of the chord of the circle $x^2+y^2=25$ intercepted on the line $x-2y=2$ is
- $\left ( \dfrac{3}{5},\dfrac{4}{5} \right )$
- $(-2,-2)$
- $\left ( \dfrac{2}{5},-\dfrac{4}{5} \right )$
- $\left ( \dfrac{8}{3},\dfrac{1}{3} \right )$
The midpoint (h, k) of a chord of x^2 + y^2 = r^2 with equation lx + my = n is given by the condition that the line joining the origin to the midpoint is perpendicular to the chord. The line x - 2y = 2 has slope 1/2. The perpendicular line through the origin has slope -2, so y = -2x. Substituting into x - 2y = 2: x - 2(-2x) = 2, 5x = 2, x = 2/5. Then y = -4/5. Midpoint is (2/5, -4/5).
The perpendicular from the center (0,0) of the circle to the chord bisects the chord. The slope of the given line x minus 2y equals 2 is 1 by 2, so the slope of the perpendicular line is negative 2, making its equation y equals negative 2x. Substituting y equals negative 2x into the line equation x minus 2(negative 2x) equals 2 gives 5x equals 2. Thus, x equals 2 by 5, and y equals negative 4 by 5.