Question

The locus of the centroid of the triangle formed by any point P on the hyperbola 16x² – 9y² + 32x + 36y – 164 = 0, and its foci is : 

(1) 16x² – 9y² + 32x + 36y – 36 = 0 

(2) 9x² – 16y² + 36x + 32y – 144 = 0 

(3) 16x² – 9y² + 32x + 36y – 144 = 0 

(4) 9x² – 16y² + 36x + 32y – 36 = 0

Answer 1

Given hyperbola is

16(x + 1)² – 9(y – 2)² = 164 + 16 – 36 = 144

Let the centroid be (h, k) 

& A(α, β) be point on hyperbola

So h = \(\frac{α-6+4}{3}\), k = \(\frac{\beta+2+2}{3}\)

\(\Rightarrow\) α = 3h + 2, β = 3k – 4 (α, β) lies on hyperbola so 

16(3h + 2 + 1)² – 9(3k – 4 – 2)² = 144 

\(\Rightarrow\) 144(h + 1)² – 81(k – 2)² = 144 

\(\Rightarrow\) 16(h² + 2h + 1) – 9(k² – 4k + 4) = 16 

\(\Rightarrow\) 16x² – 9y² + 32x + 36y – 36 = 0