Definition of Circle

Q.

Find the radius of the circle $x^2+y^2-2x+4y-11=0$

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Q. $a{x}^2+2y^2+2bxy+2x-y+c=0$ represents a circle through the origin, if

1. a = 0, b = 0, c = 2

2. a = 1, b = 0, c = 0
3. a = 2, b = 2, c = 0
4. a = 2, b = 0, c = 0

Q. If a variable circle having fixed radius $a$, passes through origin and meets the coordinates axes at point $A$ and $B$ respectively, then the locus of centroid of $△OAB$, where $O$ is the origin, is

1. $9(x^2+y^2)=4a^2$
2. $9(x^2+y^2)=a^2$
3. $9(x^2+y^2)=2a^2$
4. $9(x^2+y^2)=8a^2$

Q.

The equation of the pair of straight lines parallel to $x$ - axis and touching the circle $x^2+y^2-6x-4y-12=0$ is

1. $y^2–4y–21=0$

2. $y^2+4y–21=0$

3. $y^2–4y+21=0$

4. $y^2+4y+21=0$

Q.

The values of $\lambda$ for which the circle $x^2+y^2+6x+5+\lambda (x^2+y^2-8x+7)=0$ dwindles into a point are

1. $1\pm \frac{\sqrt{2}}{3}$
   

2. $2\pm \frac{2\sqrt{2}}{3}$
   

3. $2\pm \frac{4\sqrt{2}}{3}$
   

4. $1\pm \frac{4\sqrt{2}}{3}$

Q. The equation of circle passing through the points $(4,1),(6,5)$ and having centre on the line $4x+y=16$ is

1. $x^2+y^2-2x-24y+15=0$
2. $x^2+y^2-2x-24y-15=0$
3. $x^2+y^2-6x-8y+25=0$
4. $x^2+y^2-6x-8y+15=0$

Q. Let $x,y$ be real variables satisfying the $x^2+y^2+8x-10y-40=0$. Let $a=max\left\{\sqrt{(x+2{)}^2+(y-3{)}^2}\right\}$ and $b=min\left\{\sqrt{(x+2{)}^2+(y-3{)}^2}\right\}$, then

1. $a+b=18$
2. $a+b=\sqrt{2}$
3. $a-b=4\sqrt{2}$
4. $a.b=73$

Q.

If $(m_i,\frac{1}{m_i})$, i = 1, 2, 3, 4 are con - cyclic points, then the value of $m_1{m}_2{m}_3{m}_4$ is

1. 1

2. -1

3. 0

4. 2

Q. If the two circles $(x-1{)}^2+(y-3{)}^2=r^2$ and $x^2+y^2-8x+2y+8=0$ intersect at two distinct points, then

1. 2 < r < 8
2. r < 2
3. r = 2
4. r > 2

Q. If one of the diameters of the circle, given by the equation $x^2+y^2-4x+6y-12=0$, is a chord of a circle S, whose centre is at (-3, 2), then the radius of S is

1. $5\sqrt{2}$
2. $5\sqrt{3}$
3. $5$
4. $10$

Q. A variable circle passes through the fixed point (2, 0) and touches the y-axis . Then the locus of its centre is

1. A circle

2. An Ellipse

3. A hyperbola
4. A parabola

Q. The equation $a{x}^2+b{y}^2+2hxy+2gx+2fy+c=0$ will represent a circle, if

1. a = b = 0 and c = 0

2. f = g and h = 0
3. a = b ≠ 0 and h = 0
4. f = g and c = 0

Q. $L_1$ is a line intersecting $x$ and $y$ axes at $A(a,0)$ and $B(0,b).$ $L_2$ is a line perpendicular to $L_1$ intersecting $x$ and $y$ axes at $C$ and $D$ respectiveley. What is the condition for the common chord of the circles with $BD$ and $AC$ as diameters to pass through the point $(a,b)?$

1. $a=2b$
2. $2a=b$
3. $a=-b$
4. $a=-2b$

Q. If the point diametrically opposite to point $P(1,0)$ on the circle $x^2+y^2+2x+4y-3=0$ is $(a,b)$, then

1. $|a|=3$
2. $|b|=4$
3. $|a|=4$
4. $|b|=3$

Q. The equation of circle which touches the straight lines $2x-3y=10,2x-3y=4$ and its centre lies on $2x+3y=1,$ is

1. $(x-2{)}^2+(y+1{)}^2=\frac{36}{13}$
2. $(x-2{)}^2+(y+1{)}^2=\frac{9}{13}$
3. $(x-3{)}^2+(y+2{)}^2=\frac{9}{13}$
4. $(x-3{)}^2+(y+2{)}^2=\frac{36}{13}$

Q. The distance of the point $(1,2)$ from the centre of circle $x^2+y^2-8x+4y+16=0$ is units

Q. If the abscissa and ordinates of two points $P$ and $Q$ are the roots of the equations $x^2+2ax-b^2=0$ and $x^2+2px-q^2=0$ respectively, then the equation of the circle with $PQ$ as diameter is

Q. If complex number $z$ satisfies the inequality $|z-1-2i|\le 1$, then which of the following option(s) is (are) CORRECT ?

1. maximum value of $|z|$ is $\sqrt{5}+1$
2. minimum value of $|z|$ is $\sqrt{5}$
3. maximum value of $\arg \left(z\right)$ is $\frac{\pi }{2}$
4. minimum value of $\arg \left(z\right)$ is ${\tan }^{-1}\left(\frac{3}{4}\right)$

Q.

Two parallel chords of a circle of radius 2 units are at a distance $\sqrt{3}+1$ units apart. If the chords subtend at the centre, angles of $\frac{\pi }{k}$ and $\frac{2\pi }{k}$, where $k>0$, then the value of [k] (where [k] denotes the largest integer less than or equal to k) is ___

Q. If the line x + 2by + 7 = 0 is diameter of the circle $x^2+y^2-6x+2y=0$, then b =

1. 3
2. -5
3. -1
4. 5

Q. The centre of the circle x = 2 + 3 cos $\theta$, y = 3 sin $\theta$ - 1 is

1. (3, 3)
2. (2, -1)
3. (-2, 1)
4. (-1, 2)

Q.

Let $\varphi (x,y)=0$ be the equation of a circle, If $\varphi (0,\lambda )=0$ has equal roots $\lambda =2,2$ and $\varphi (\lambda ,0)=0$ has roots $\lambda =\frac{4}{5},5,$ then the roots of the circle is

1. (2, 29/100)

2. (29/10, 2)
   

3. (-2, 29/10)
   

4. none of these

Q. The equation of circle whose two diameters are $2x-3y=5,$ $3x-2y=10$ and having perimeter $6\pi \text{cm},$ is

1. $(x-1{)}^2+(y+4{)}^2=9$
2. $(x+1{)}^2+(y-4{)}^2=9$
3. $(x-4{)}^2+(y-1{)}^2=9$
4. $(x+4{)}^2+(y+1{)}^2=9$

Q. Let $z$ be an imaginary complex number satisfying $|z-1|=1.$ If $\alpha =2z,$ $\beta =2\alpha$ and $\gamma =2\beta ,$ then the value of $|z{|}^2+|\alpha {|}^2+|\beta {|}^2+|\gamma {|}^2+|z-2{|}^2+|\alpha -4{|}^2+|\beta -8{|}^2+|\gamma -16{|}^2$ is

1. $100$
2. $320$
3. $340$
4. $400$

Q. If the line $3x+2\lambda y+12=0$ is a diameter of the circle $x^2+y^2-4x+6y+12=0,$ then the value of $\lambda$ is

Q. The number of points with integral coordinates that are interior to the circle $x^2+y^2=16$ is

1. 43
2. 49
3. 45
4. 51

Q. If the radius of the circle $x^2+y^2+2\lambda x-2\lambda y+18=0$ cannot exceed $6$, then the number of integral value(s) of $\lambda$ is

Q. If a variable circle having fixed radius $a$, passes through origin and meets the coordinates axes at point $A$ and $B$ respectively, then the locus of centroid of $△OAB$, where $O$ is the origin, is

Q. The centre of the circle $x=-1+2cos\theta ,y=3+2sin\theta ,$ is

1. (1, -3)
2. (-1, 3)
3. (1, 3)
4. None of these

Q. The area of the circle whose centre is at (1, 2) and which passes through the point (4, 6) is

1. $5\pi$
2. $10\pi$
3. $25\pi$
4. None of these