KCET · Maths · Sequences and Series
If \(A M\) and \(G M\) of roots of a quadratic equation are 5 and 4 , respectively, then the quadratic equation is
- A \(x^2-10 x-16=0\)
- B \(x^2+10 x+16=0\)
- C \(x^2+10 x-16=0\)
- D \(x^2-10 x+16=0\)
Answer & Solution
Correct Answer
(D) \(x^2-10 x+16=0\)
Step-by-step Solution
Detailed explanation
Let \(a\) and \(b\) be the roots the quadratic equation.
Then, the quadratic equation is
\(x^2-(a+b) x+a b=0\) ....(i)
It is given that \(\mathrm{AM}=5\) and \(\mathrm{GM}=4\)
i.e. \(\frac{a+b}{2}=5\) and \(\sqrt{a b}=4\)
\(\Rightarrow a+b=10\) and \(a b=16\)
On substituting these values in Eq. (i), we obtain. Hence, \(x^2-10 x+16=0\) is the required equation.
Then, the quadratic equation is
\(x^2-(a+b) x+a b=0\) ....(i)
It is given that \(\mathrm{AM}=5\) and \(\mathrm{GM}=4\)
i.e. \(\frac{a+b}{2}=5\) and \(\sqrt{a b}=4\)
\(\Rightarrow a+b=10\) and \(a b=16\)
On substituting these values in Eq. (i), we obtain. Hence, \(x^2-10 x+16=0\) is the required equation.
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