Question

The locus of the mid-points of the chords of an ellipse $x^2+4y^2=4$ that are drawn from the positive end of the minor axis, is

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

Answer 1

The correct option is C an ellipse with centre $(0,\frac{1}{2})$, major axis $1$ and minor axis $\frac{1}{2}$

The positive end of minor axis is $(0,1)$

Let the other end of the chord be $(h,k)$

Since point $(h,k)$ lies on the ellipse , we get $h=2cos\theta$ and $k=sin\theta$

Therefore the midpoint of chord is $(p,q)=(cos\theta ,\frac{1+sin\theta }{2})$

$\Rightarrow cos\theta =p,sin\theta =2q-1$

So we get $p^2+{(2q-1)}^2=1$

$\Rightarrow x^2+\frac{{(q-\frac{1}{2})}^2}{\frac{1}{4}}=1$

Therefore the locus is ellipse with center $(0,\frac{1}{2})$ , major axis $1$ and minor axis $\frac{1}{2}$

So the correct option is $C$