Question

The locus of mid-point of family of the chords $\lambda x+y-5=0,(\lambda$ is a parameter) of the parabola $x^2=20y$ is

• A. $y^2=10(x-5)$
• B. $x^2=10(y-5)$
• C. $x^2+y^2=25$
• D. none of these

Answer 1

Answer: (B)

Since $\lambda x+y-5=0$ passes through $(0,5)$, it is focal chord of parabola $x^2=20y$.
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 $t_1{t}_2=-1$
$\therefore k=5\left(\frac{t_1^2+t_2^2}{2}\right)=5\left(\frac{{\left(t_1+t_2\right)}^2-2t_1{t}_2}{2}\right)$
$=5\left(\frac{\frac{h^2}{25}+2}{2}\right)$
$\therefore$ Required locus is $x^2+50=10k$
or $x^2=10(y-5)$.