Question

Find the equation of the locus of a moving point P such that its distance from the point Q(2,6) and R(4,0) are in the ratio 2:1.

a) x^2 + y^2 - 10x - 12y + 40 = 0
b) x^2 + y^2 - 8x - 6y + 20 = 0
c) x^2 + y^2 - 12x - 10y + 60 = 0
d) x^2 + y^2 - 6x - 8y + 10 = 0

Answer 1

The equation of the locus is x^2 + y^2 - 8x - 6y + 20 = 0.

Step-by-step explanation:

To find the equation of the locus of a moving point P, we will use the concept of ratio of distances. Let P(x, y) be the coordinates of the moving point. The distance from P to Q(2, 6) is given by √[(x - 2)^2 + (y - 6)^2]. The distance from P to R(4, 0) is given by √[(x - 4)^2 + y^2]. According to the given condition, these distances are in the ratio 2:1, so we can write the equation:

√[(x - 2)^2 + (y - 6)^2] : √[(x - 4)^2 + y^2] = 2:1

Simplifying this equation and rearranging, we get:
x^2 + y^2 - 8x - 6y + 20 = 0