Ellipse
Locus of centre
Grade 11
Question:
<p>An ellipse has semi-major axis of length 2 and semi-minor axis of length 1. It slides between the co-ordinate axes in the first quadrant, while maintaining contact with both x-axis and y-axis. The locus of the centre of ellipse is:</p>
<p>(a) \(x^2 + y^2 = 3\)</p>
<p>(b) \(x^2 + y^2 = 5\)</p>
<p>(c) \((x - 2)^2 + (y - 1)^2 = 5\)</p>
<p>(d) \((x - 2)^2 + (y - 1)^2 = 3\)</p>
Step-by-Step Solution
Key Concept: When an ellipse with fixed semi-axes slides while maintaining contact with both coordinate axes, the distance from origin to its centre remains constant.
<p>Since the ellipse maintains contact with both coordinate axes in the first quadrant, the centre of the ellipse is at distance equal to the semi-major axis from the x-axis and semi-minor axis from the y-axis. Therefore, if the centre is at (h, k), then \(h = 2\) and \(k = 1\) when the ellipse touches both axes. As the ellipse slides while maintaining contact with both axes, the locus satisfies \(h^2 + k^2 = 4 + 1 = 5\). Thus the locus is \(x^2 + y^2 = 5\).</p>
Correct Answer: B