First slide

Theory of equations

Question

If $(m_{r}, 1 / m_{r}), r = 1, 2, 3, 4$ , are four pairs of values of x and y that satisfy the equation $x^{2} + y^{2} + 2 gx + 2 fy + c = 0$ , then the value of $m_{1} m_{2} m_{3} m_{4}$ is

Moderate

A

0
B

1
C

-1
D

none of these

Solution

If $[m_{r}, (1 / m_{r})]$ satisfy the given equation
$x^{2} + y^{2} + 2 gx + 2 fy + c = 0$ , then
$\begin{aligned} m_{r}^{2} + \frac{1}{m_{r}^{2}} + 2 {gm}_{r} + \frac{2 f}{m_{r}} + c = 0 \\ \Rightarrow m_{r}^{4} + 2 {gm}_{r}^{3} + {cm}_{r}^{2} + 2 {fm}_{r} + 1 = 0 \end{aligned}$
Now, roots of given equation are $m_{1}, m_{2}, m_{3}, m_{4}$ . The product of roots
$m_{1} m_{2} m_{3} m_{4} = \frac{Constant term}{Coefficient of m_{r}^{4}} = \frac{1}{1} = 1$

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