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Q.
The locus of mid-point of family of the chords $\lambda x+y-5=0,\left(\lambda\right.$ is a parameter) of the parabola $x^{2}=20 y$ is
Conic Sections
Solution:
Since $\lambda x+y-5=0$ passes through $(0,5)$, it is focal chord of parabola $x^{2}=20 y$.
Let $(h, k)$ be the mid-point of all such chords
$\therefore h=5\left(t_{1}+t_{2}\right), k=5\left(\frac{t_{1}^{2}+t_{2}^{2}}{2}\right)$
Also $t_{1} t_{2}=-1$
$\therefore k=5\left(\frac{t_{1}^{2}+t_{2}^{2}}{2}\right)=5\left(\frac{\left(t_{1}+t_{2}\right)^{2}-2 t_{1} t_{2}}{2}\right)$
$=5\left(\frac{\frac{h^{2}}{25}+2}{2}\right)$
$\therefore $ Required locus is $x^{2}+50=10 k$
or $x^{2}=10(y-5)$.