Question

A stick of length $10$ units rests against the floor and a wall of a room. If the stick begins to slide on the floor then the locus of its middle point is
(Assume the origin to be the intersection of the floor and the wall and the coordinate axes along the floor and the wall)

• A. $x^2+y^2=50$
• B. $x^2+y^2=25$
• C. $x^2+y^2=100$
• D. None of these

Answer 1

The correct option is B $x^2+y^2=25$

Let the end points of stick be $A(a,0)$ and $B(0,b)$ with equation $\frac{x}{a}+\frac{y}{b}=1$
whose length is $10$
i.e $AB=\sqrt{a^2+b^2}=10$
Let mid point of $AB$ is $(x_1,y_1)$
$\Rightarrow x_1=\frac{a}{2}$ and $y_1=\frac{b}{2}$
$\Rightarrow 4x_1^2+4y_1^2=100$
Hence, the locus of the mid point is $x^2+y^2=25$