First slide

Theory of equations

Question

$If the system of equations r^{2} + s^{2} = t and r + s + t = \frac{k - 3}{2} has exactly one real solution, then the value of k is$

Moderate

A

1
B

2
C

3
D

4

Solution

Eliminating t, we get
$\begin{array}{l} r + s + r^{2} + s^{2} = \frac{k - 3}{2} \\ or r^{2} + r + s^{2} + s + \frac{3 - k}{2} = 0 \end{array}$
Now for one real solution of $r$
D = 0
$\begin{array}{l} \Rightarrow 1 - 4 (s^{2} + s + \frac{3 - k}{2}) = 0 \\ \Rightarrow 4 s^{2} + 4 s + 5 - 2 k = 0 \\ Now for one real solution ' s' \\ D = 0 \\ \Rightarrow 16 - 16 (5 - 2 k) = 0 \\ \Rightarrow 1 = 5 - 2 k \\ \Rightarrow k = 2 \end{array}$

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