Three distinct points are given in the 2-dimensional coordinate plane such that the ratio of the distance of any one of them from the point (1,0) to the distance from the point (-1,0) is equal to 1/3. Then the locus of point A is?

Let the point A, P, and Q be (h,k), (1,0), and (-1,0) respectively.
AP/AQ = 1/3 ⇒ 3AP = AQ ⇒ 9AP² = AQ² ⇒ 9[(h+1)² + k²] = (h-1)² + k²
⇒ 9(h² + 2h + 1 + k²) = h² - 2h + 1 + k²
⇒ 9h² + 18h + 9 + 9k² = h² - 2h + 1 + k²
⇒ 8h² + 20h + 8k² + 8 = 0
⇒ h² + (5/2)h + k² + 1 = 0
Locus of A(h,k) is x² + y² + (5/2)x + 1 = 0