Ellipse
Locus and definition of ellipse
Grade 11
Question:
<p>Let O(0, 0) and A(0, 1) be two fixed points, then the locus of a point P such that the perimeter of △AOP is 4, is</p>
<p>(a) \(8x^2 - 9y^2 + 9y = 18\)</p>
<p>(b) \(9x^2 - 8y^2 + 8y = 16\)</p>
<p>(c) \(9x^2 + 8y^2 - 8y = 16\)</p>
<p>(d) \(8x^2 + 9y^2 - 9y = 18\)</p>
Step-by-Step Solution
Key Concept: The locus of a point where the sum of distances from two fixed points is constant is an ellipse. Apply the ellipse definition and simplify.
<p><strong>Step 1:</strong> Given vertices of △AOP are O(0, 0) and A(0, 1).</p><p><strong>Step 2:</strong> Let the coordinates of point P be $(x, y)$.</p><p><strong>Step 3:</strong> The perimeter of △AOP is given by $OA + AP + PO = 4$.</p><p><strong>Step 4:</strong> We have $OA = 1$.</p><p><strong>Step 5:</strong> Therefore, $AP + PO = 3$.</p><p><strong>Step 6:</strong> This represents an ellipse with foci at O(0, 0) and A(0, 1), where the sum of distances equals 3.</p><p><strong>Step 7:</strong> Using the standard form of ellipse, we get $9x^2 + 8y^2 - 8y = 16$.</p><p>∴ Answer is (c).</p>
Correct Answer: c