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  • The locus of the mid-point of the line segment joining the focus to a moving point on the parabola  <img alt="y^{2}=4ax" src="https://entrancecorner.oncodecogs.com/gif.latex?y%5

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola  y^{2}=4ax   is another parabola with directrix 

Option: 1

x=-a


Option: 2

x=-\frac{a}{2}


Option: 3

x=0


Option: 4

x=\frac{a}{2}


Answers (1)

best_answer

Let P(h,k)  be the midpoint of the line segment joining the focus (a,0) and a general point  Q(x,y)  on the parabola. Then,

h=\frac{x+a}{2}\, ,k=\frac{y}{2}

\begin{aligned} \Rightarrow & x=2h-a,y=2k \end{aligned}

Put these values of x and  y and in y^{2}=4ax 

4k^{2}=4a(2h-a)\\\

 \begin{aligned} \Rightarrow & 4k^{2}=8ah-4a^{2}\\ \Rightarrow & k^2=2ah-2a^{2} \end{aligned}

So , P(h,k)  locus is

\begin{aligned} & y^2=2 a x-2 a^2 \\ & y^2=2 a x-2 a^2 \\ \Rightarrow & y^2=2 a\left(x-\frac{a}{2}\right) \end{aligned}

Its directrix is x-\frac{a}{2}=-\frac{a}{2}\Rightarrow x=0

 

Posted by

Sanket Gandhi

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