Question:
<p>If one end of a focal chord AB of the parabola y<sup>2</sup> = 8x is at <span class="math-tex">\(A\left(\frac{1}{\sqrt{2}},-2\right)\)</span>, then the equation of the tangent to it at B is:</p>
<p style="display:inline">x + 2y + 8 = 0</p>
<p style="display:inline">x - 2y + 8 = 0</p>
<p style="display:inline">2x - y - 24 = 0</p>
<p style="display:inline">2x + y - 24 = 0</p>
Step-by-Step Solution
Key Concept: The endpoints of a focal chord of the parabola $y^2 = 4ax$ have parameters $t_1$ and $t_2$ such that $t_1t_2 = -1$, allowing the calculation of point B's coordinates and its tangent equation.
<p><img alt="" src="https://media-mycbseguide.s3.amazonaws.com/images/imgur/1621087726-x6at9n.jpg" style="height:172px; width:150px" /><br />
Let parabola y<sup>2</sup> = 8x at point <span class="math-tex">$\left(\frac{1}{2},-2\right)$</span> is (2t<sup>2</sup>, 4t)<br />
<span class="math-tex">$\Rightarrow t=\frac{-1}{2}$</span><br />
Parameter of other end of focal chord is 2<br />
So, coordinates of B is (8, 8)<br />
<span class="math-tex">$\Rightarrow$</span> Equation of tangent at B is 8y - 4(x + 8) = 0<br />
<span class="math-tex">$\Rightarrow$</span> 2y - x = 8<br />
<span class="math-tex">$\Rightarrow$</span> x - 2y + 8 = 0</p>
Correct Answer: B