Circles
Tangent Lines and Distance
Grade 11
Question:
<p>Let C be the circle of radius unity centred at the origin. If two positive numbers \(x_1\) and \(x_2\) are such that the line passing through \((x_1, -1)\) and \((x_2, 1)\) is tangent to C, then:</p>
<p>(a) \(x_1 x_2 = 1\)</p>
<p>(b) \(x_1 x_2 = -1\)</p>
<p>(c) \(x_1 + x_2 = 1\)</p>
<p>(d) \(4x_1 x_2 = 1\)</p>
Step-by-Step Solution
Key Concept: Use the condition that the distance from the center of a circle to a tangent line equals the radius; apply the point-to-line distance formula.
<p>The line passing through \((x_1, -1)\) and \((x_2, 1)\) has slope \(\frac{1-(-1)}{x_2-x_1} = \frac{2}{x_2-x_1}\). The equation of the line is: \(y + 1 = \frac{2}{x_2-x_1}(x - x_1)\), which simplifies to \(2x - (x_2-x_1)y - 2x_1 - (x_2-x_1) = 0\). For this line to be tangent to the circle \(x^2 + y^2 = 1\), the distance from the origin to the line must equal 1. Using the distance formula: \(\frac{|−2x_1 − (x_2−x_1)|}{ \sqrt{4+(x_2−x_1)^2}} = 1\). Simplifying and solving yields \(x_1 x_2 = 1\).</p>
Correct Answer: A