Explanation

(c)
Let two points on the parabola are $p({at}_{1}^{2},2{at}_{1})$ and $\mathcal{Q}\ ({at}_{2}^{2},2{at}_{2})$
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Now, $SP = \sqrt{\left( a - {at}_{1}^{2} \right)^{2}} + \left( 0 - 2at_{1} \right)^{2}$
$= a + \ {at}_{1}^{2}$
$S\mathcal{Q}\ = \sqrt{\left( a - {at}_{2}^{2} \right)^{2}} + \left( 0 - 2{at}_{2} \right)^{2}$
$= \ a + {at}_{2}^{2} = a + \frac{a}{t_{1}^{2}}\ \ \ \ \ \left( \because t_{1}t_{2} = - 1 \right)$
Now, $\frac{2 \times SP \times S\mathcal{Q}}{SP + S\mathcal{Q}} = \frac{2 \times a(1 + t_{1}^{2}) \times a\left( 1 + \frac{1}{t_{1}^{2}} \right)}{\left( a + {at}_{1}^{2} \right) + \left( a + \frac{a}{t_{1}^{2}} \right)}$
$= \frac{2a\left( 2 + \frac{1}{t_{1}^{2}} + t_{1}^{2} \right)}{\left( 2 + \frac{1}{t_{1}^{2}} + t_{1}^{2} \right)} = 2a = l\ (given)$
Hence, $SP,\ l,\ S\mathcal{Q}\$are in HP