Question

If a variable circle x² + y² - 2ax + 4ay = 0 intersects the hyperbola xy = 4 at the points (x_i, y_i) = 1, 2, 3, 4 then locus of the point $\left( \frac{x_{1} + x_{2} + x_{3} + x_{4}}{4},\frac{y_{1} + y_{2} + y_{3} + y_{4}}{4} \right)$ is:

• A. y + 2x = 0
• B. y - 2x + 5 = 0
• C. y - 2x = 0
• D. y + 4x - 7 = 0

Answer 1

Answer: (A)

y + 2x = 0

Answer 2

Putting y = 4/x in the equation of the circle, we get. x⁴ - 2ax³ + 16ax + 16 = 0.

So $\frac{x_{1} + x_{2} + x_{3} + x_{4}}{4} = \frac{a}{2}$

Similarly$\frac{y_{1} + y_{2} + y_{3} + y_{4}}{4} = - a$.

⇒ h = a/2, k = -a

⇒ h = -k/2 ⇒ y + 2x = 0.