Question

Let $AB$ be a line segment of length $4$ units with $A$ on line $y=2x$ and $B$ on the line $y=x,$ the locus of middle points of all such line segments is a

• A. Parabola
• B. Ellipse
• C. pair of straight lines
• D. circle

Answer 1

The correct option is B Ellipse

As given that $A$ lies on line $y=2x$ and $B$ on the line $y=x$
Let $A(t,2t)$ and $B(k,k);AB=4$
$\sqrt{(t-k{)}^2+(2t-k{)}^2}=4$
$\Rightarrow (t-k{)}^2+(2t-k{)}^2=16\dots \left(1\right)$

and let mid point of $AB$ be $(x,y)$
$\frac{t+k}{2}=x,\frac{k+2t}{2}=y$
$\Rightarrow t=2(y-x),k=4x-2y$
By equation $\left(1\right)$
$(4y-6x{)}^2+(6y-8x{)}^2=16$
$\Rightarrow (2y-3x{)}^2+(3y-4x{)}^2=4$
$\Rightarrow 25x^2+13y^2-36xy-4=0$
$\Rightarrow △=\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|$
$\Rightarrow △=\left|\begin{array}{ccc}25& -18& 0\\ -18& 13& 0\\ 0& 0& -4\end{array}\right|$
$\Rightarrow △\ne 0$
and
$h^2<ab$ So, it's an ellipse