Question:
<p>The equation of the tangent to the parabola y<sup>2</sup> = 16x, which is perpendicular to the line y = 3x + 7 is</p>
<p style="display:inline">3y - x + 36 = 0</p>
<p style="display:inline">3y + x + 36 = 0</p>
<p style="display:inline">y - 3x + 4 = 0</p>
<p style="display:inline">3y + x - 36 = 0</p>
Step-by-Step Solution
Key Concept: Determine the slope of the tangent using the perpendicularity rule and then apply the tangency condition c = a/m for the parabola y^2 = 4ax.
<p>Line perpendicular to given line is, 3<span class="math-tex">$\lambda$</span> + x = <span class="math-tex">$\lambda$</span><br />
<span class="math-tex">$\therefore y=\frac{-1}{3} x+\frac{\lambda}{3}$</span><br />
Here, <span class="math-tex">$m=\frac{-1}{3}, c=\frac{\lambda}{3}$</span><br />
If we compare y<sup>2</sup> = 16 x with y<sup>2</sup> = 4ax then a = 4<br />
Condition for tangency is,<br />
<span class="math-tex">$c=\frac{a}{m} \Rightarrow \frac{\lambda}{3}=\frac{4}{(-1 / 3)}$</span><br />
<span class="math-tex">$\Rightarrow \lambda$</span> = -36<br />
Required equation is x + 3y + 36 = 0.</p>
Correct Answer: B