Q188 / 270 · NDA 2019 (Phase 2) · Mathematics

Consider the following statements in respect to the quadratic equation $4(x-p)(x-q)-r^2=0$ where $p, q$ and $r$ are real numbers : 1. The roots are real 2. The roots are equal if $p=q$ and $r=0$ Which of the above statements is/are correct?

  • A
    1 only
  • B
    2 only
  • Correct answer
    C
    Both 1 and 2
  • D
    Neither 1 nor 2

Solution

$\begin{aligned} & \text { } 4(x-p)(x-q)-r^2=0 \\ & =(x-p)(x-q)-\frac{r^2}{4}=0 \\ & =x^2-(p+q) x+p q-\frac{r^2}{4}=0 \end{aligned}$ Discriminat, $\begin{aligned} & \mathrm{D}=[-(p+q)]^2-4\left(p q-\frac{r^2}{4}\right)=0 \\ & \mathrm{D}=(p+q)^2-4 p q+r^2=0 \\ & \mathrm{D}=(p-q)^2+r^2=0 \end{aligned}$ We have $\mathrm{D} \geq 0 \Rightarrow$ real roots. and if $p=q, r=0$ the $\mathrm{D}=0$ $\therefore$ equal roots $\therefore$ both statements are correct.

Hence, the correct answer is option: 'C'.

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