Key points to solve the problem:

• Idea of distance formula- Distance between two points A(x₁,y₁) and B(x₂,y₂) is given by- AB =

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 point whose locus is to be determined to be (h,k)

Distance of (h,k) from (2,0) =

Distance of (h,k) from (1,3) =

According to the question:

Squaring both sides:

16{(h - 2)² + k² } = 25{(h - 1)² + (k - 3)² }

⇒ 16{h² + 4 - 4h + k² } = 25{h² - 2h + 1 + k² - 6k + 9}

⇒ 9h² + 9k² + 14h - 150k + 186 = 0

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

Thus, the locus of a point which moves such that the ratio of its distance from (2, 0) and (1, 3) is 5 : 4 is –

9x² + 9y² + 14x - 150y + 186 = 0 ….ans