Circles
Locus and tangent properties
Grade 11
Question:
<p>Let l₁, l₂ and l₃ are the lengths of the tangents drawn from a variable point P to the circle x² + y² = a², x² + y² = 2ax and x² + y² = 2ay respectively. The lengths satisfy the relation l₁⁴ = l₂²l₃² + a⁴. Then the locus of P can be:</p>
<p>(a) Line</p>
<p>(b) Circle</p>
<p>(c) Parabola</p>
<p>(d) Hyperbola</p>
Step-by-Step Solution
Key Concept: Use the tangent length formula and the given algebraic relation to derive the locus equation.
<p>For a point P(h, k), the length of tangent to a circle x² + y² = r² is √(h² + k² - r²). Using this formula for all three circles and substituting into the given relation l₁⁴ = l₂²l₃² + a⁴ yields the locus equation.</p>
Correct Answer: A, B