Let \(P(x, y)\) be the mid point of the line joining \((1,0)\) to a point on the curve \(\mathrm{y}^{2}=\left|\begin{array}{ll}\mathrm{x}+1 & \mathrm{x}+2 \\ \mathrm{x}+3 & \mathrm{x}+5\end{array}\right|\). Then, locus of \(\mathrm{P}\) is symmetrical about

The given curve \(\mathrm{y}^{2}=\left|\begin{array}{ll}\mathrm{x}+1 & \mathrm{x}+2 \\ \mathrm{x}+3 & \mathrm{x}+5\end{array}\right|\)



\(
\begin{aligned}
&\Rightarrow \quad \mathrm{y}^{2}=(\mathrm{x}+1)(\mathrm{x}+5)-(\mathrm{x}+2)(\mathrm{x}+3) \\
&\Rightarrow \quad \mathrm{y}^{2}=\mathrm{x}^{2}+6 \mathrm{x}+5-\mathrm{x}^{2}-5 \mathrm{x}-6 \\
&\quad \mathrm{y}^{2}=\mathrm{x}-1 \text { which is parabola. }
\end{aligned}
\)
Let the parametric coordinate of parabola is
\(
\mathrm{Q}=\left(\frac{\mathrm{t}^{2}}{4}, \frac{\mathrm{t}}{2}+1\right)
\)
Now, given that \(\mathrm{P}(\mathrm{x}, \mathrm{y})\) is the mid point of \(\mathrm{AQ}\).
\(
P(x, y)=\left[\frac{\frac{t^{2}}{4}+1}{2}, \frac{\frac{t}{2}+1}{2}\right]
\)
\(
\begin{array}{ll}
\Rightarrow & \mathrm{x}=\frac{\mathrm{t}^{2}}{8}+\frac{1}{2} \Rightarrow 8 \mathrm{x}=\mathrm{t}^{2}+4 \\
\text { and } & \mathrm{y}=\frac{\mathrm{t}}{4}+\frac{1}{2} \Rightarrow \mathrm{t}=2(2 \mathrm{y}-1) \\
\Rightarrow & 8 \mathrm{x}=4(2 \mathrm{y}-1)^{2}+4 \\
\Rightarrow & 2 \mathrm{x}=(2 \mathrm{y}-1)^{2}+1 \\
\Rightarrow & (2 \mathrm{y}-1)^{2}=(2 \mathrm{x}-1) \\
\Rightarrow & 4\left(\mathrm{y}-\frac{1}{2}\right)^{2}=2\left(\mathrm{x}-\frac{1}{2}\right) \\
\Rightarrow & \left(\mathrm{y}-\frac{1}{2}\right)^{2}=\frac{1}{2}\left(\mathrm{x}-\frac{1}{2}\right)
\end{array}
\)
Which is also a parabola.
Hence, the locus of \(\mathrm{P}(\mathrm{x}, \mathrm{y})\) is symmetrical about \(x\)-axis ie, \(\mathrm{y}=0\) from figure.