Question

The locus of the midpoints of the chords of an ellpse $x^2+4y^2=4$ that are drawn form the positive end of the minor axis, is

• A. a circle with centre $\left(\frac{1}{2},0\right)$ and radius 1
• B. a parabola with focus $\left(\frac{1}{2},0\right)$ and directrix $x=-1$
• C. an ellipse with centre $\left(0,\frac{1}{2}\right)$ major axis 1 and minor axis $\frac{1}{2}$
• D. a hyperbola with centre $\left(0,\frac{1}{2}\right)$ transverse axis 1 and conjugate axis $\frac{1}{2}$

Answer 1

Answer: (C)

Given equation of an ellipse is
$x^2+4y^2=4$
$\Rightarrow \frac{x^2}{4}+\frac{y^2}{1}=1$
$\therefore$ Coordinate of positive end of minor axis is $B(0,1)$. Let mid-point of the chord $BP$ is $M(h,k)$
Then, $(h,k)=\left(\frac{0+x}{2},\frac{1+y}{2}\right)$
$\Rightarrow h=\frac{x}{2}\Rightarrow x=2h$
and $k=\frac{1+y}{2}\Rightarrow y=2k-1$
$\therefore P(x,y)\equiv \{2h,(2k-1)\}$
Since, the point ' $P$ ' lies cllipes so form Eq. (i), we get
$(2h{)}_1^2+4(2k-1{)}^2=4$
$\Rightarrow 4h^2+4(2k-1{)}^2=4$
$\Rightarrow h^2+4k^2+1-4k=1$
$\Rightarrow h^2+4k^2-4k=0$
Thus, required locus is an ellipse whose equation is $x^2+4y^2-4y=0$
$\Rightarrow \frac{(x-0{)}^2}{1}+\frac{{\left(y-\frac{1}{2}\right)}^2}{\left(\frac{1}{4}\right)}=1$
whose centre $\left(0,\frac{1}{2}\right)$ and major and minor axis $\frac{1}{2}$