Question

If the circle \( x^{2} + y^{2} = a^{2} \) intersects the hyperbola \( xy = c^{2} \) in four points \( (x_i, y_i) \), for \( i = 1, 2, 3, 4 \), then \( y_{1} + y_{2} + y_{3} + y_{4} \) equals

• A. 0
• B. c
• C. a
• D. $c^{4}$

Hint:

In a quartic equation, if the $y^3$ term is missing, the sum of all roots is zero.

Answer 1

The Correct Option is A

Step 1: Form the Equation
Substitute $x = c^{2}/y$ into the circle equation $x^{2}+y^{2}=a^{2}$. $(c^{2}/y)^{2} + y^{2} = a^{2} \Rightarrow \frac{c^{4}}{y^{2}} + y^{2} = a^{2}$.
Step 2: Simplify to Polynomial
$c^{4} + y^{4} = a^{2}y^{2} \Rightarrow y^{4} - a^{2}y^{2} + c^{4} = 0$.
Step 3: Sum of Roots
For the quartic equation $y^{4} + 0y^{3} - a^{2}y^{2} + 0y + c^{4} = 0$, the sum of roots $y_{1}+y_{2}+y_{3}+y_{4}$ is the negative of the coefficient of $y^{3}$. Sum $= 0$.
Final Answer: (a)