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

Option: 1

\mathrm{y}^{2}=10(\mathrm{x}-5)


Option: 2

\mathrm{x}^{2}=10(\mathrm{y}-5)


Option: 3

\mathrm{x^{2}+y^{2}=25}


Option: 4

none of these


Answers (1)

best_answer

\mathrm{\lambda x+y-5=0} is the focal chord

Since it passes through (0,5)

\mathrm{Let \, \mathrm{h}, \mathrm{k}} be the mid point of all such chords

\mathrm{\mathrm{h}=5\left(\mathrm{t}_{1}+\mathrm{t}_{2}\right), \mathrm{k}=5\left(\frac{\mathrm{t}_{1}^{2}+\mathrm{t}_{2}^{2}}{2}\right)\left(\mathrm{t}_{1} \mathrm{t}_{2}-1\right)}
On elimianting \mathrm{\mathrm{t}_{1}, \mathrm{t}_{2}}

\mathrm{\mathrm{x}^{2}=10(\mathrm{y}-5)}

Posted by

Divya Prakash Singh

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