Question

If a circle makes intercepts of length 5 and 3 on two perpendicular lines, then the locus of the centre of the circle is

• A. A parabola
• B. An ellipse
• C. A hyperbola
• D. None of these

Answer 1

Answer: (C)

A hyperbola

Answer 2

Let the two given ⊥ lines be the coordinate axes and let the equation of variable circle be

x² + y² + 2gx + 2fy + c = 0 ... (1)

Then, 5 = 2$\sqrt{g^{2} - c}$ and 3 = 2$\sqrt{f^{2} - c}$.

Squaring and subtracting these, we get

4(g² - c) - 4 (f² - c) = 25 - 9.

⇒ g² - f² = 4 or (- g)² - (-f)² = 4.

Hence, locus of the centre (-g, -f) of circle is x² - y² = 4, which is a rectangular hyperbola.