Consider the following statements:I. The inte | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Statement $I$: For the circle $x^2+y^2-2x-4y+1=0$,the intercept on the $Y$-axis is given by $2\sqrt{f^2-c}$,where $f = -2$ and $c = 1$. Thus,the i

Vedclass · Accepted Answer

Vedclass · Answer

(C) Statement $I$: For the circle $x^2+y^2-2x-4y+1=0$,the intercept on the $Y$-axis is given by $2\sqrt{f^2-c}$,where $f = -2$ and $c = 1$. Thus,the intercept is $2\sqrt{(-2)^2-1} = 2\sqrt{4-1} = 2\sqrt{3}$. So,statement $I$ is True.Statement $II$: For the circle $x^2+y^2-4x-2y+6=0$,the intercept on the $X$-axis is given by $2\sqrt{g^2-c}$,where $g = -2$ and $c = 6$. Thus,the intercept is $2\sqrt{(-2)^2-6} = 2\sqrt{4-6} = 2\sqrt{-2}$. Since the value inside the square root is negative,the circle does not intersect the $X$-axis. So,statement $II$ is False.Statement $III$: The line $y=2x+1$ or $2x-y+1=0$ intersects the circle $x^2+y^2=9$ at two distinct points if the perpendicular distance $p$ from the center $(0,0)$ to the line is less than the radius $r=3$. Here,$p = \frac{|2(0)-(0)+1|}{\sqrt{2^2+(-1)^2}} = \frac{1}{\sqrt{5}}$. Since $p = \frac{1}{\sqrt{5}} Therefore,$I$ is True,$II$ is False,and $III$ is True. The correct option is $(c)$.

Similar Questions

Let the normals at all the points on a given curve pass through a fixed point $(a, b)$. If the curve passes through $(3, -3)$ and $(4, -2\sqrt{2})$,and given that $a - 2\sqrt{2}b = 3$,then $(a^{2} + b^{2} + ab)$ is equal to ..... .

If two distinct chords,drawn from the point $(p, q)$ on the circle ${x^2} + {y^2} = px + qy$ (where $pq \neq 0$),are bisected by the $x$-axis,then:

$A$ rectangle is inscribed in a circle with a diameter lying along the line $3y = x + 7$. If the two adjacent vertices of the rectangle are $(-8, 5)$ and $(6, 5)$,then the area of the rectangle (in $sq. units$) is

Let a circle $C$ touch the lines $L_{1}: 4x - 3y + K_{1} = 0$ and $L_{2}: 4x - 3y + K_{2} = 0$,where $K_{1}, K_{2} \in R$. If a line passing through the centre of the circle $C$ intersects $L_{1}$ at $(-1, 2)$ and $L_{2}$ at $(3, -6)$,then the equation of the circle $C$ is:

The minute hand of a watch is $1.5 \, \text{cm}$ long. How far does its tip move in $40$ minutes (in $, \text{cm}$)? (Use $\pi = 3.14$).

Vedclass Test Series

Exam Paper Generator

Online Exam Module