Question:
<p>If the parabola y<sup>2</sup> = 4ax passes through (-3, 2), then the length of its lams rectum is</p>
<p style="display:inline"><span class="math-tex">\(\frac{4}{3}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{2}{3}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{1}{3}\)</span></p>
<p style="display:inline">4</p>
Step-by-Step Solution
Key Concept: Substitute the coordinates of the given point into the parabola's equation to solve for the absolute value of 4a, which represents the length of the latus rectum.
<p>Parabola y<sup>2</sup> = 4ax has vertex (0, 0) and given that the curve passes through (-3, 2) (which is a point of 2<sup>nd</sup> quadrant)<br />
<span class="math-tex">$\Rightarrow$</span> Parabola opens on left of y-axis <span class="math-tex">$\Rightarrow$</span> a < 0.<br />
Take a = -a' where a' > 0<br />
<span class="math-tex">$\therefore$</span> The equation of parabola is y<sup>2</sup> = -4a'x<br />
The point (-3, 2) will satisfy the equation y<sup>2</sup> = -4a'x<br />
<span class="math-tex">$\Rightarrow 4 a^{\prime}=\frac{4}{3}$</span><br />
<span class="math-tex">$\Rightarrow$</span> The length of its latus rectum is <span class="math-tex">$\frac{4}{3}$</span>.</p>
Correct Answer: A