Circles
Tangent to Circle
Grade 11
Question:
<p>Equation of line that touches the curves |<em>y</em>| = <em>x</em><sup>2</sup> and <em>x</em><sup>2</sup> + (<em>y</em> - 2)<sup>2</sup> = 4 where <em>x</em> ≠ 0 is:</p>
<p>(a) <em>y</em> = 4√5<em>x</em> + 20</p>
<p>(b) <em>y</em> = 4√3<em>x</em> - 12</p>
<p>(c) <em>y</em> = -4√5<em>x</em> + 20</p>
<p>(d) <em>y</em> = -4√5<em>x</em> - 20</p>
Step-by-Step Solution
Key Concept: A common tangent to both curves must satisfy tangency conditions for the parabola and the circle simultaneously.
<p><strong>Solution:</strong> A line tangent to both curves must satisfy the tangency condition for each curve. For the parabola |<em>y</em>| = <em>x</em><sup>2</sup>, the tangent line at point (<em>t</em>, <em>t</em><sup>2</sup>) has the form <em>y</em> = 2<em>tx</em> - <em>t</em><sup>2</sup>. For this line to be tangent to the circle <em>x</em><sup>2</sup> + (<em>y</em> - 2)<sup>2</sup> = 4 (center at (0, 2) with radius 2), the distance from the center to the line must equal 2. Solving these conditions simultaneously yields the three tangent lines.</p>
Correct Answer: a, b, c