Quadratic Equations
Common root condition
nta_pyq_2023_jan
Grade 11
Question:
If the value of real number $a > 0$ for which $x^2 - 5ax + 1 = 0$ and $x^2 - ax - 5 = 0$ have a common real root is $\dfrac{3}{\sqrt{2\beta}}$, then $\beta$ is equal to ______.
Step-by-Step Solution
Key Concept: For a common root $r$: subtracting the two equations eliminates $x^2$ and gives $r$ in terms of $a$. Then substitute back.
Subtracting: $-4ax + 6 = 0 \Rightarrow x = \frac{3}{2a}$. Common root condition: $(4a)(26a) = 36 \Rightarrow a^2 = \frac{9}{26} \Rightarrow a = \frac{3}{\sqrt{26}}$. Given $a = \frac{3}{\sqrt{2\beta}}$, so $2\beta = 26 \Rightarrow \beta = 13$.
Correct Answer: 13