stick of lantern units rest against the floor and a Wall of room if the sticks begins to start on floor than the locus of its middle

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 :

Answer 1

Locus is the set of points that satisfy a certain property. Now, we have to determine the property of the midpoint of the stick when it starts sliding. Now we consider the floor as the x-axis and wall as the y-axis. 

The stick in the sliding position is drawn below.

Now AB is the stick that is sliding and P is its midpoint. As point A touches the floor, it has y – coordinate as 0. Similarly, point B touches the wall, it has x – coordinate as 0. Now, P is the midpoint of stick AB. By midpoint formula, if there are two points E (a, b) and F (c, d), then its midpoint will be given by,

Now, we will apply the Pythagoras theorem in the right-angled triangle AOB. According to Pythagoras theorem, we can say that if H is the hypotenuse, B is the base and P is the perpendicular, then,

H² = P² + B²

In our case, H = AB, P = OB and B = OA. Thus, we have,

Now, we will put the values of x₁ and y₁ from (iv) and (v) to (iii). Thus, we will get,

On putting h = x and k = y, we will get,

⇒ x² + y² = 25