Question

Consider the following statements.
I. The intercept made by the circle $x^2+y^2-2x-4y+1=0$ on $Y$ -axis is $2\sqrt{3}$
II. The intercept made by the circle $x^2+y^2-4x-2y+6=0$ on $X$ -axis is $2\sqrt{2}$
III. The straight line $y=2x+1$ 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) $x^2+y^2-2x-4y+1=0$ intercept on
$Y$ -axis is $2\sqrt{f^2-c}$
$\Rightarrow 2\sqrt{4-1}$
$\Rightarrow 2\sqrt{3}$ True
(II) $x^2+y^2-4x-2y-6=0$ intercept on $X$ -axis $2\sqrt{4+6}$
$\Rightarrow 2\sqrt{10}\ne 2\sqrt{2}$ False
(III) Line $y=2x+1$ 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.