Key points to solve the problem:

• Idea of section formula- Let two points A(x₁,y₁) and B(x₂,y₂) forms a line segment. If a point C(x,y) divides line segment AB in the ratio of m:n internally, then coordinates of C is given as:

C = when m = n =1 , C becomes the midpoint of AB and C is given as C =

How to approach: To find the locus of a point we first assume the coordinate of the point to be (h, k) and write a mathematical equation as per the conditions mentioned in the question and finally replace (h, k) with (x, y) to get the locus of the point.

Let the coordinates of a point whose locus is to be determined to be (h, k). Name the moving point to be C

As, coordinate of mid point is (h,k) {by our assumption},

Let Q(a,b) be the point such that Q lies on curve y² = x

b² = a ……equation 1

According to question C is the midpoint of OQ

∵ C = ⇒ C =

∴

Similarly,

Putting values of a and b in equation 1,we have:

Replace (h,k) with (x,y)

Thus, the locus of a point is: ….ans