Q87 / 160 · AP EAMCET 2022 (06 Jul Shift 2) · Mathematics

If the circle $x^2+y^2+2 \alpha x+c=0$ lies completely inside the circle $x^2+y^2+2 \beta x+c=0$, then which of the following holds?

  • A
    $\alpha \beta < 0$
  • B
    $c < 0$
  • C
    $c=0$
  • Correct answer
    D
    $\alpha \beta>0$

Solution

The centre of the circle $x^2+y^2+2 \alpha x+c=0$ is $(-\alpha, 0)$. As the circle $x^2+y^2+2 \alpha x+c=0$ lies inside the circle $x^2+y^2+2 \beta x+c=0$, then $$ \begin{aligned} & (-\alpha)^2+0^2+2 \beta(-\alpha)+c < 0 \\ & \alpha^2-2 \alpha \beta+c < 0 \\ & \alpha^2-2 \alpha \beta+\beta^2 < \beta^2-c \\ & (\alpha-\beta)^2 < \beta^2-c \end{aligned} $$ Also, radius of circle $x^2+y^2+2 \alpha x+c=0$ less than radius of the circle $x^2+y^2+2 \beta x+c=0$ $$ \begin{aligned} & \sqrt{\alpha^2-c} < \sqrt{\beta^2-c} \\ & \Rightarrow \alpha^2-c < \beta^2-c \Rightarrow \alpha < \beta \\ & \alpha^2-2 \alpha \beta < 0 \\ & \end{aligned} $$ From Eqs. (i) and (ii), $$ \begin{aligned} & c>0 \\ & \Rightarrow \quad-2 \alpha \beta < 0 \\ & \alpha \beta>0 \\ & \end{aligned} $$

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

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