<p>The centroid of the triangle formed by joining the feet of the normals drawn from any point to the parabola y<sup>2</sup> = 4ax, lies on the</p>
<p style="display:inline">directrix</p>
<p style="display:inline">axis</p>
<p style="display:inline">latus rectum</p>
<p style="display:inline">tangent at vertex</p>
Step-by-Step Solution
Key Concept: The centroid of the feet of co-normal points lies on the parabola's axis because the sum of their ordinates is zero, a result of the cubic equation for the slope m lacking a quadratic term.
<p>Normal to parabola is given by<br />
y = mx - 2am - am<sup>3</sup> which is a cubic equation in m.<br />
<span class="math-tex">$\Rightarrow$</span> Three normals can be drawn on parabola<br />
y<sup>2</sup> = 4ax. Since coefficient of m<sup>2</sup> = 0<br />
<span class="math-tex">$\Rightarrow$</span> m<sub>1</sub> + m<sub>2</sub> + m<sub>3</sub> = 0, If (x<sub>1</sub>, y<sub>1</sub>), (x<sub>2</sub>, y<sub>2</sub>), (x<sub>3</sub>, y<sub>3</sub>)<br />
are feet of the normals then <span class="math-tex">$m_{1}=-\frac{y_{1}}{2 a}$</span>, <span class="math-tex">$m_{2}=-\frac{y_{2}}{2 a}, m_{3}=-\frac{y_{3}}{2 a}$</span><br />
<span class="math-tex">$\Rightarrow$</span> y<sub>1</sub> + y<sub>2</sub> + y<sub>3</sub> = 0<br />
<span class="math-tex">$\Rightarrow$</span> y coordinate of the centroid of the triangle, with feet of the normals as vertices, is zero.<br />
So centroid lies on the axis of the parabola.</p>
Correct Answer: B