Circles
Chords of Contact
Grade 11
Question:
<p>The distance between the chords of contact of tangents to the circle \(x^2 + y^2 + 2gx + 2fy + c = 0\) from the origin and the point \((g, f)\) is:</p>
<p>(a) \(\sqrt{g^2 + f^2}\)</p>
<p>(b) \(\frac{g^2 + f^2 - c}{\sqrt{g^2 + f^2}}\)</p>
<p>(c) \(\frac{g^2 + f^2 - c}{2\sqrt{g^2 + f^2}}\)</p>
<p>(d) \(\frac{g^2 + f^2 - c}{2(g^2 + f^2)}\)</p>
Step-by-Step Solution
Key Concept: Find the equations of both chords of contact and apply the distance formula for parallel lines.
<p>The circle has center \(C = (-g, -f)\) and radius \(r = \sqrt{g^2 + f^2 - c}\). The chord of contact from the origin is given by: \(gx + fy + c = 0\). The chord of contact from point \((g, f)\) is obtained by replacing \((x, y)\) with \((g, f)\) in the tangent from external point formula. The distance between two parallel chords (both chords of contact are perpendicular to the line joining their respective external points) is found using the distance between parallel lines formula. After calculation, the distance is \(\frac{g^2 + f^2 - c}{2\sqrt{g^2 + f^2}}\).</p>
Correct Answer: C