Question

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?

I	II	III
(a)	True	True	True
(b)	True	True	False
(c)	True	False	True
(d)	False	False	True

• A. a
• B. b
• C. c
• D. d

Answer 1

Answer: (C)

(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 \, r^{2}=9 $
$ \Rightarrow \, p=\frac{1}{\sqrt{5}}$
$\Rightarrow \, p^{2}=\frac{1}{5}$
$ \Rightarrow \,9-\frac{1}{5}>\,0$ is it is True.