Circles
Chord Properties
Grade 11
Question:
<p>If a line having y-intercept \('c'\) makes a chord of length \('a'\) to the circle \(x^2 + y^2 = a^2\), then:</p>
<p>(a) \(4c^2 \geq 3a^2\)</p>
<p>(b) \(4c^2 < 3a^2\)</p>
<p>(c) \(4c^2 > 5a^2\)</p>
<p>(d) \(c^2 > 3a^2\)</p>
Step-by-Step Solution
Key Concept: Use the chord length formula with the perpendicular distance from the center to establish a relationship between the y-intercept and the radius.
<p>Let the line be \(y = mx + c\). The distance from center \((0,0)\) to the line is \(d = \frac{|c|}{\sqrt{1+m^2}}\). For a chord of length \(a\) in circle \(x^2 + y^2 = a^2\) (radius = a), we have: \(d^2 + \left(\frac{a}{2}\right)^2 = a^2\), so \(d^2 = \frac{3a^2}{4}\). Thus \(\frac{c^2}{1+m^2} = \frac{3a^2}{4}\). Since \(m^2 \geq 0\), we have \(1 + m^2 \geq 1\), so \(c^2 = \frac{3a^2(1+m^2)}{4} \geq \frac{3a^2}{4}\). Therefore \(4c^2 \geq 3a^2\).</p>
Correct Answer: A