Ellipse
Tangents and Latus Rectum
Grade 11
Question:
<p>If ellipses \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) are described having the same major axis, but a variable minor axis, then for all values of \(b\), the tangents at the ends of their latus rectum can pass through which point?</p>
<p>(a) \((0, a)\)</p>
<p>(b) \((0, -a)\)</p>
<p>(c) \((0, 2a)\)</p>
<p>(d) \((0, -2a)\)</p>
Step-by-Step Solution
Key Concept: The tangent to an ellipse at the end of latus rectum has a special property: all such tangents (as \(b\) varies) pass through fixed points on the minor axis.
<p>For ellipses with fixed major axis \(a\) and variable minor axis \(b\), the ends of the latus rectum are at \(\left(\pm c, \pm\frac{b^2}{a}\right)\) where \(c = \sqrt{a^2 - b^2}\).</p><p>The equation of the tangent at \(\left(c, \frac{b^2}{a}\right)\) is: \(\frac{cx}{a^2} + \frac{b^2y}{ab^2} = 1\), which simplifies to \(\frac{cx}{a^2} + \frac{y}{a} = 1\).</p><p>Setting \(x = 0\): \(y = a\). Similarly, for other ends, we get \(y = \pm a\). Thus tangents pass through \((0, a)\) and \((0, -a)\).</p>
Correct Answer: a, b