Question:
<p>If lx + my + n = 0 is tangent to the parabola x<sup>2</sup> = y, then condition of tangency is</p>
<p style="display:inline">l = 4m<sup>2</sup>n<sup>2</sup></p>
<p style="display:inline">m<sup>2</sup> = 4ln</p>
<p style="display:inline">l<sup>2</sup> = 4mn</p>
<p style="display:inline">l<sup>2</sup> = 2mn</p>
Step-by-Step Solution
Key Concept: The condition for a line to be tangent to a parabola is that the system of equations must have exactly one solution, which occurs when the discriminant of the resulting quadratic equation is zero.
<p>We have, lx + my + n = 0<br />
<span class="math-tex">$\Rightarrow x=\frac{-m}{l} y-\frac{n}{l}$</span> ...(i)<br />
and x<sup>2</sup> = y ...(ii)<br />
Now, condition of tangency is <span class="math-tex">$c=\frac{a}{m}$</span><br />
<span class="math-tex">$\Rightarrow \frac{-n}{l}=\frac{1}{4} \times\left(\frac{-l}{m}\right)$</span> ...[From (i) and (ii)]<br />
<span class="math-tex">$\Rightarrow$</span> l<sup>2</sup> = 4 mn</p>
Correct Answer: C