Question

If the mid-point of the chord intercepted by the circle \(x^2 + y^2 - 8x + 10y + 5 = 0\) on the line \(2x + y + 2 = 0\) is \((h, k)\), then \(k + 4h =\) ?

• A. \(2\)
• B. \(0\)
• C. \(1\)
• D. \(-1\)

Hint:

To find the midpoint of a chord intercepted by a circle on a line, substitute line into circle, solve the resulting quadratic, and average the roots.

Answer 1

The Correct Option is A

To find the midpoint of the chord cut by a circle on a line, substitute the line equation in the circle to get a quadratic. The midpoint is the average of the roots' coordinates. Circle: \(x^2 + y^2 - 8x + 10y + 5 = 0\) Line: \(2x + y + 2 = 0 \Rightarrow y = -2x - 2\) Substitute into the circle: \[ x^2 + (-2x - 2)^2 - 8x + 10(-2x - 2) + 5 = 0 \] \[ x^2 + 4x^2 + 8x + 4 - 8x - 20x - 20 + 5 = 0 \Rightarrow 5x^2 - 20x - 11 = 0 \] Let roots be \(x_1, x_2\), then: \[ \text{Midpoint } h = \frac{x_1 + x_2}{2} = \frac{20}{10} = 2,\quad y = -2h - 2 = -6 \Rightarrow k = -6 \] So, \(k + 4h = -6 + 8 = 2\)