A circle cuts a chord of length on the -axis and passes through a point on the -axis, distant from the origin. Then, the locus of the centre of this circle, is
- A.a parabola
- B.an ellipse
- C.a straight line
- D.a hyperbola
Show answer and solution
Correct option: A
Solution
Let the center of the circle be (h, k). The circle cuts a chord of length 4a on the X-axis, so the half-chord is 2a, meaning the distance from center to X-axis satisfies: radius² = k² + (2a)² = k² + 4a².
The circle passes through (0, 2b), so radius² = h² + (k - 2b)².
Setting equal: k² + 4a² = h² + k² - 4bk + 4b²
This simplifies to: h² = 4bk - 4b² + 4a² = 4b·k - 4(b² - a²)
Rewriting: h² = 4b·k - 4(b² - a²), i.e., x² = 4b·y - 4(b² - a²), which is x² = 4b(y - (b² - a²)/b).
This is of the form X² = 4bY, a parabola.
The circle passes through (0, 2b), so radius² = h² + (k - 2b)².
Setting equal: k² + 4a² = h² + k² - 4bk + 4b²
This simplifies to: h² = 4bk - 4b² + 4a² = 4b·k - 4(b² - a²)
Rewriting: h² = 4b·k - 4(b² - a²), i.e., x² = 4b·y - 4(b² - a²), which is x² = 4b(y - (b² - a²)/b).
This is of the form X² = 4bY, a parabola.
[JEE Main 2019 · 11 Jan · Shift II]