Circles
Chord of Contact of Tangents
Grade 11
Question:
<p>Let the tangents drawn from the origin to the circle <math>x^2 + y^2 - 8x - 4y + 16 = 0</math> touch it at the points A and B. The <math>(AB)^2</math> is equal to</p>
<p>(a) <math>\frac{56}{5}</math></p>
<p>(b) <math>\frac{52}{5}</math></p>
<p>(c) <math>\frac{64}{5}</math></p>
<p>(d) <math>\frac{32}{5}</math></p>
Step-by-Step Solution
Key Concept: Use the chord of contact formula to find the equation of chord AB, then calculate its length using the distance from center to chord and the radius.
<p><strong>Solution:</strong> The equation of chord of contact AB to circle <math>x^2 + y^2 - 8x - 4y + 16 = 0</math> w.r.t. point origin <math>(0, 0)</math> is <math>T = 0</math></p><p>Using the chord of contact formula: <math>xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0</math></p><p>Substituting <math>(x_1, y_1) = (0, 0)</math>, <math>g = -4</math>, <math>f = -2</math>, <math>c = 16</math>:</p><p><math>0 + 0 - 4x - 2y + 16 = 0</math></p><p><math>2x + y = 8</math></p><p>Finding the chord length using the distance formula from center <math>(4, 2)</math> to the line, and using the radius, we get <math>(AB)^2 = \frac{64}{5}</math></p><p>∴ Answer is (c).</p>
Correct Answer: C