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Question: If the equation \(\left( a-5 \right){{x}^{2}}+2\left( a-10 \right)x+a+10=0\) has roots of opposite s...

If the equation (a−5)x2+2(a−10)x+a+10=0\left( a-5 \right){{x}^{2}}+2\left( a-10 \right)x+a+10=0 has roots of opposite sign, then,
A. a>10a> 10
B. −15<a<15-15< a <15
C. −10<a<5-10< a< 5
D. None of these.

Explanation

Solution

Assume one root to be α\alpha , therefore the other root will be −α-\alpha . Now we will apply the formula of sum of roots of a quadratic equation [−ba]\left[ \dfrac{-b}{a} \right] to solve further and get the desired answer.

Complete step-by-step solution:
Let us assume one of the roots to be α\alpha , therefore we will get the other root as −α-\alpha as we are given that the equation has roots of opposite sign. Now we also know that,
Sum of roots of a quadratic equation = [−ba]\left[ \dfrac{-b}{a} \right], where - b is the coefficient of x and a is the coefficient of x2{{x}^{2}}. Therefore we can say that in the given equation, (a−5)x2+2(a−10)x+a+10=0\left( a-5 \right){{x}^{2}}+2\left( a-10 \right)x+a+10=0, the coefficient of x=2(a−10)x=2\left( a-10 \right) and the coefficient of x2=(a−5){{x}^{2}}=\left( a-5 \right). Now, we know that the,
sum of roots = - coefficient of xcoefficient of x2\text{sum of roots = }\dfrac{\text{- coefficient of x}}{\text{coefficient of }{{\text{x}}^{\text{2}}}}
So, by substituting the values of coefficient of x and coefficient of x2{{x}^{2}}, we will get,
α+(−α)=−2(a−10)(a−5) ⇒α−α=−2(a−10)(a−5) ⇒0=−2a+20a−5 \begin{aligned} & \alpha +\left( -\alpha \right)=\dfrac{-2\left( a-10 \right)}{\left( a-5 \right)} \\\ & \Rightarrow \alpha -\alpha =\dfrac{-2\left( a-10 \right)}{\left( a-5 \right)} \\\ & \Rightarrow 0=\dfrac{-2a+20}{a-5} \\\ \end{aligned}
We will now transfer (a−5)\left( a-5 \right) to the left hand side or the LHS using cross multiplication. So, we get,

& -2a+20=0 \\\ & \Rightarrow 2a=20 \\\ & \Rightarrow a=\dfrac{20}{2} \\\ & \Rightarrow a=10 \\\ \end{aligned}$$ Therefore, when we check the options that are provide to us, we find that the most suitable one is $-15< a < 15$. **Hence, the correct answer is option B.** **Note:** The students should not make mistakes by getting confused with the formula of sum of roots and product of roots, which is $\dfrac{c}{a}$, where c is the constant and a is the coefficient of the term ${{x}^{2}}$ in the quadratic equation. The students should also watch their calculations as according to the standard equation, $a{{x}^{2}}+bx+c$ and in the given equation, we have a as the common variable, so don’t get confused by that.