Question

A ladder starts sliding on a wall from its initial position (fig 1) and (fig 2) is an intermediate stage
before it completely hits the ground. Find the locus of the curve traced by the mid-point of the ladder.
(a and b are constants).

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

We will start with the intermediate stage given

Let P(h,k) be the coordinates of the centre. We have to find a relation between h and k without variables. Here a and b are constant. So it is not a problem if they appear in the final equation.

We know that the length of the ladder remains the same. This will be the condition that we will use to find the locus. This is not given in the question. So the key step in this problem is coming up with this condition.

Length of the ladder =$\sqrt{a^2+b^2}$
(Its the hypotenuse in the above figure)
Let us consider the position of the ladder at some other time.

[P(h,k) is a variable point. Don't think of it as a constant point. The centre of the ladder actually moves]
Here $\sqrt{x_1^2+y_1^2}=\sqrt{a^2+b^2}$= lenght of the ladder
$\Rightarrow x_1^2+y_1^2=a^2+b^2\to \left(1\right)$

Is this the equation of our locus? No, x and y are not the coordinates of the mid-point. $(\frac{x_1+0}{2},\frac{0+y_1}{2})=(h,k)$ = centre of ladder.
$\Rightarrow x_1=2hand{y}_1=2k-\left(2\right)$
$\left(1\right),\left(2\right)\Rightarrow 4h^2+4k^2=a^2+b^2$
Or $h^2+k^2=\frac{a^2+b^2}{4}$
We can replace (h,k) with (x,y) because change of variable does not change the equation
$\Rightarrow x^2+y^2=\frac{a^2+b^2}{4}$