Hyperbola
Tangent Condition
Grade 11
Question:
<p>The line <i>x</i> cos <i>α</i> + <i>y</i> sin <i>α</i> = <i>p</i> touches the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), if <i>a</i><sup>2</sup> cos<sup>2</sup> <i>α</i> − <i>b</i><sup>2</sup> sin<sup>2</sup> <i>α</i> is equal to</p>
<p>(a) <i>p</i></p>
<p>(b) <i>p</i><sup>2</sup></p>
<p>(c) −<i>p</i><sup>2</sup></p>
<p>(d) 2<i>p</i></p>
Step-by-Step Solution
Key Concept: For a line to be tangent to a hyperbola, the condition c² = a²m² − b² must be satisfied. Use the given line equation and compare coefficients.
<p><strong>Given:</strong> The line <i>x</i> cos <i>α</i> + <i>y</i> sin <i>α</i> = <i>p</i></p><p><strong>Rewrite as:</strong> $y = -x \cot \alpha + p \csc \alpha$</p><p><strong>Compare with:</strong> $y = mx + c$, we get $m = -\cot \alpha$ and $c = p \csc \alpha$</p><p><strong>Condition for tangency:</strong> $c^2 = a^2m^2 - b^2$</p><p><strong>Substitute:</strong> $p^2 \csc^2 \alpha = a^2 \cot^2 \alpha - b^2$</p><p><strong>Simplify:</strong> $a^2 \cos^2 \alpha - b^2 \sin^2 \alpha = p^2$</p><p>∴ Answer is <strong>(b) p²</strong></p>
Correct Answer: B