Q89 / 160 · TS EAMCET 2018 (04 May Shift 2) · Mathematics

Consider the following statements. I. The intercept made by the circle $x^2+y^2-2 x-4 y+1=0$ on $Y$-axis is $2 \sqrt{3}$ II. The intercept made by the circle $x^2+y^2-4 x-2 y+6=0$ on $X$-axis is $2 \sqrt{2}$ III. The straight line $y=2 x+1$ cuts the circle $x^2+y^2=9$ at two distinct points Then which one of the following options is correct? Figure

  • A
    (a)
  • B
    (b)
  • Correct answer
    C
    (c)
  • D
    (d)

Solution

(I) $x^2+y^2-2 x-4 y+1=0$ intercept on $Y$-axis is $2 \sqrt{f^2-c} \Rightarrow 2 \sqrt{4-1} \Rightarrow 2 \sqrt{3}$ True (II) $x^2+y^2-4 x-2 y-6=0$ intercept on $X$-axis $2 \sqrt{4+6} \Rightarrow 2 \sqrt{10} \neq 2 \sqrt{2}$ False (III) Line $y=2 x+1$ at the circle $x^2+y^2-9=0$ So, $r^2-p^2>0$ $\Rightarrow \quad r^2=9 \Rightarrow p=\frac{1}{\sqrt{5}}$ $\Rightarrow \quad p^2=\frac{1}{5} \Rightarrow 9-\frac{1}{5}>0$ is it is True.

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

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