Ellipse
Chord properties
Grade 11
Question:
<p>If <\tan\theta_1\tan\theta_2 = -\frac{a^2}{b^2}>, then the chord joining two points <\theta_1> and <\theta_2> on the ellipse <\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1> will subtend a right angle at</p>
<p>(a) focus</p>
<p>(b) centre</p>
<p>(c) end of the major axis</p>
<p>(d) end of minor axis</p>
Step-by-Step Solution
Key Concept: The condition for a chord to subtend a right angle at a point is that the dot product of position vectors equals zero. Using parametric form on the ellipse, this condition directly yields the given relation.
<p><strong>Solution:</strong></p><p>Let <\P(a\cos\theta_1, b\sin\theta_1)> and <\Q(a\cos\theta_2, b\sin\theta_2)> be two points on the ellipse.</p><p>For the chord PQ to subtend a right angle at the centre O, we need:</p><p><\vec{OP} \cdot \vec{OQ} = 0></p><p>This gives: <\a^2\cos\theta_1\cos\theta_2 + b^2\sin\theta_1\sin\theta_2 = 0></p><p>Dividing by <\cos\theta_1\cos\theta_2>:</p><p><\a^2 + b^2\tan\theta_1\tan\theta_2 = 0></p><p><\tan\theta_1\tan\theta_2 = -\frac{a^2}{b^2}></p><p>∴ The chord subtends a right angle at the <strong>centre</strong>.</p>
Correct Answer: B