Question

A circle passes through \( (0, a) \) and \( (b, h) \), and its center is at \( (c, 0) \). Find \( c \).

• A. \( \frac{b^2 - a^2 + h^2}{2b} \)
• B. \( \frac{b^2 + a^2 - h^2}{2b} \)
• C. \( \frac{b^2 - a^2 + h^2}{2a} \)
• D. \( \frac{b^2 + a^2 - h^2}{2a} \)

Hint:

Use the distance formula from center to two known points and equate them since both lie on the circle.

Answer 1

The Correct Option is A

Let center = \( (c, 0) \) Radius = distance from center to each point. From point \( (0, a) \): \[ % Option (c)^2 + a^2 = r^2 \] From point \( (b, h) \): \[ (b - c)^2 + h^2 = r^2 \] Equating: \[ \begin{align} c^2 + a^2 = (b - c)^2 + h^2 \Rightarrow c^2 + a^2 = b^2 - 2bc + c^2 + h^2 \Rightarrow a^2 = b^2 - 2bc + h^2 \Rightarrow 2bc = b^2 + h^2 - a^2 \Rightarrow c = \frac{b^2 - a^2 + h^2}{2b} \]