Question

If $I$ denotes the semi - latus rectum of the parabola$y^2=4ax$ and $SP$ and $SQ$ denote the segments of any focal chord $PQ,S$ being the focus, then$SP,I$ and $SQ$ are in the relation.

• A. $AP$
• B. $GP$
• C. $HP$
• D. $I^2=S{P}^2+S{Q}^2$

Answer 1

The correct option is C

$HP$

Explanation for correct option:

Find the relation:

Let us assume 2 points on the parabola $P(a{t}_1^2,2a{t}_1)$ and$Q=(a{t}_2^2,2a{t}_2)$

Then,

$t_1{t}_2=-1$ $\left[\because \text{P and Q pass via focus}\right]$

$$\begin{gathered} SP=\sqrt{{(a–a{t}_1^2)}^2+{(0–2a{t}_1)}^2} \\ =a+a{t}_1^2 \\ SQ=\sqrt{{(a–a{t}_2^2)}^2+{(0–2a{t}_2)}^2} \\ =a+a{t}_2^2 \\ =a+\left(\frac{a}{{t_1}^2}\right) \end{gathered}$$

Now,

$\begin{array}{rcl}\frac{1}{SP}+\frac{1}{SQ}& =& \frac{1}{a+a{t}_1^2}+\frac{1}{a+{\displaystyle \frac{a}{t_1^2}}}\\ & \Rightarrow & \frac{SP+SQ}{SP·SQ}=\frac{1}{a}\left[\frac{t_1^2+1}{t_1^2+1}\right]\\ & \Rightarrow & \frac{SP+SQ}{SP·SQ}=\frac{1}{a}\\ & \Rightarrow & \frac{2SP·SQ}{SP+SQ}=2a\left[\text{multiply by 2 on both side}\right]\\ & =& I\end{array}$

Hence, $SP,I,SQ$ are in HP.

Hence, the correct option is C.