Circles
Tangents to Circles
Grade 11
Question:
<p>Let L₁, L₂ and L₃ be the lengths of tangents drawn from a point P to the circles x² + y² = 4, x² + y² - 4x = 0 and x² + y² - 4y = 0 respectively. If L₁⁴ = L₂² L₃² + 16 then the locus of P are the curves, C₁ (a straight line) and C₂ (a circle).</p><p>Circumcentre of the triangle formed by C₁ and two other lines which are at angle of 45° with C₁ and tangent to C₂ is:</p>
<p>(a) (1, 1)</p>
<p>(b) (0, 0)</p>
<p>(c) (-1, -1)</p>
<p>(d) (2, 2)</p>
Step-by-Step Solution
Key Concept: Use the tangent length formula and the given condition to find the locus, then use geometric properties of the configuration to locate the circumcentre.
<p>From the condition L₁⁴ = L₂² L₃² + 16, we derive the locus as a combination of a line C₁ and a circle C₂. The tangent lines at 45° to C₁ that are also tangent to C₂ form a triangle with C₁. The circumcentre of this triangle lies at the origin by symmetry.</p>
Correct Answer: B