Circles
Tangent to Circle
Grade 11
Question:
<p>Let <em>l</em><sub>1</sub>, <em>l</em><sub>2</sub> and <em>l</em><sub>3</sub> are the lengths of the tangents drawn from a variable point P to the circle <em>x</em><sup>2</sup> + <em>y</em><sup>2</sup> = <em>a</em><sup>2</sup>; <em>x</em><sup>2</sup> + <em>y</em><sup>2</sup> = 2<em>ax</em> and <em>x</em><sup>2</sup> + <em>y</em><sup>2</sup> = 2<em>ay</em> respectively. The lengths satisfy the relation <em>l</em><sub>1</sub><sup>4</sup> = <em>l</em><sub>2</sub><sup>2</sup><em>l</em><sub>3</sub><sup>2</sup> + <em>a</em><sup>4</sup>. 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: Express the tangent lengths in terms of coordinates and apply the given relation to find the locus.
<p><strong>Solution:</strong> Let P be at coordinates (<em>x</em>, <em>y</em>). The lengths of tangents from P to the three circles are:<br/><em>l</em><sub>1</sub><sup>2</sup> = <em>x</em><sup>2</sup> + <em>y</em><sup>2</sup> - <em>a</em><sup>2</sup><br/><em>l</em><sub>2</sub><sup>2</sup> = <em>x</em><sup>2</sup> + <em>y</em><sup>2</sup> - 2<em>ax</em><br/><em>l</em><sub>3</sub><sup>2</sup> = <em>x</em><sup>2</sup> + <em>y</em><sup>2</sup> - 2<em>ay</em><br/>Substituting into the given relation <em>l</em><sub>1</sub><sup>4</sup> = <em>l</em><sub>2</sub><sup>2</sup><em>l</em><sub>3</sub><sup>2</sup> + <em>a</em><sup>4</sup> and simplifying yields equations representing both a line and a circle as loci.</p>
Correct Answer: a, b