If \(3 x^2-7 x+2=0\) and \(15 x^2-11 x+a=0\) have a common root and \(a\) is a positive real number, then…
If \(3 x^2-7 x+2=0\) and \(15 x^2-11 x+a=0\) have a common root and \(a\) is a positive real number, then the sum of the roots of the equation \(15 x^2-a x+7=0\), is
\(\frac{76}{15}\)
\(\frac{38}{15}\)
\(\frac{2}{15}\)
\(\frac{36}{15}\)
Solution
Given equations are
\(\text { and } \begin{aligned}
3 x^2-7 x+2 & =0 \\
15 x^2-11 x+a & =0 \\
\text { Here, } \quad a_1=3, b_1=-7, c_1 & =2 \\
a_2=15, b_2=-11, c_2 & =a
\end{aligned}\)
and
Let \(\alpha\) is a common root of the Eqs. (i) and (ii), Then, \(\alpha\) will satisfy both the equations.
\(\therefore\) Common root is given by
\(\begin{aligned}
& (2 \times 15-a \times 3)^2=(-7 a+22(-33+105) \\
& \Rightarrow \quad(30-3 a)^2=(22-7 a)(72 \\
& \Rightarrow \quad\left[9(10-a)^2\right]=(22-7 a)(72 \\
& \Rightarrow \quad(10-a)^2=(22-7 a) 8 \\
& \Rightarrow 100+a^2-20 a=176-56 a \\
& \Rightarrow \quad a^2+36 a-76=0 \\
& \Rightarrow \quad a^2+38 a-2 a-76=0 \\
& \Rightarrow \quad a(a+38)-2(a+38)=0 \\
& \Rightarrow \quad(a+38)(a-2)=0 \\
& \Rightarrow \quad a=2 \quad[\because a > 0]
\end{aligned}\)
Now, for equation \(15 x^2-a x+7=0\)
\(\begin{aligned}
\text { Sum of roots } & =\frac{- \text { Coefficient of } x}{\text { Coefficient of } x^2} \\
& =\frac{-(-a)}{15}=\frac{a}{15}=\frac{2}{15}
\end{aligned}\)