First slide

Theory of equations

Question

$If (m_{r}, \frac{1}{m_{r}}); r = 1, 2, 3, 4 are four pairs of values of x and y that satisfy the equation$
$x^{2} + y^{2} + 2 g x + 2 f y + c = 0, then value of m_{1} \cdot m_{2} \cdot m_{3} \cdot m_{4}$

Moderate

A

0
B

1
C

-1
D

None of these

Solution

$(m_{r}, \frac{1}{m_{r}}) satisfy the given equation x^{2} + y^{2} + 2 g x + 2 f y + c = 0$
$then m_{r}^{2} + \frac{1}{m_{r}^{2}} + 2 g m_{r} + \frac{2 f}{m_{r}} + c = 0$
$\Rightarrow m_{r}^{4} + 2 g m_{r}^{3} + c m_{r}^{2} + 2 f m_{r} + 1 = 0$
$Now roots of given equation are m_{1}, m_{2}, m_{3}, m_{4}$
$Product of roots = m_{1} m_{2} m_{3} m_{4}$
$= \frac{constant term}{coefficeint of m_{r}^{4}} = \frac{1}{1} = 1$

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