Question:
<p>The line 3x - 4y = 5 is a tangent to the hyperbola x<sup>2</sup> - 4y<sup>2</sup> = 5. The point of contact is</p>
<p style="display:inline">(3, 1)</p>
<p style="display:inline"><span class="math-tex">\(\left(\frac{1}{4}\right),2)\)</span></p>
<p style="display:inline"><span class="math-tex">\(\left(2, \frac{1}{4}\right)\)</span></p>
<p style="display:inline">(1, 3)</p>
Step-by-Step Solution
Key Concept: The point of contact for a tangent line $y = mx + c$ to a standard hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is calculated using the formula $(-\frac{a^2m}{c}, -\frac{b^2}{c})$.
<p>We have <span class="math-tex">$y=\frac{3}{4} x-\frac{5}{4}$</span> and <span class="math-tex">$\frac{x^{2}}{5}-\frac{y^{2}}{\frac{5}{4}}=1$</span><br />
Here <span class="math-tex">$m=\frac{3}{4}$</span>, a<sup>2</sup> = 5 and <span class="math-tex">$b^{2}=\frac{5}{4}$</span><br />
The point of contact is <span class="math-tex">$\left(\frac{\pm 5\left(\frac{3}{4}\right)}{\sqrt{5 \times \frac{9}{16}-\frac{5}{4}}}, \frac{\pm \frac{5}{4}}{\sqrt{5 \times \frac{9}{16}-\frac{5}{4}}}\right)$</span><br />
<span class="math-tex">$\Rightarrow$</span> (3, 1) is the required point.</p>
Correct Answer: A