Definition of Circle

Q. If $z$ is a complex number such that $\frac{z-i}{z-1}$ is purely imaginary, then the minimum value of $|z–(3+3i)|$ is

1. $6\sqrt{2}$
2. $2\sqrt{2}$
3. $3\sqrt{2}$
4. $2\sqrt{2}-1$

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

Q.

The Number Of Integral Values Of Lambda For Which $X^2+Y^2+\lambda X+(1-\lambda )$ $Y+5=0$ Is The Equation Of A Circle Whose Radius Cannot Exceed $5$, Is

1. $14$

2. $16$

3. $18$

4. $20$

Q. If for the complex number $z$ satisfying $|z-2-2i|\le 1,$ the maximum value of $|3iz+6|$ is attained at $a+ib$, then $a+b$ is equal to

Q. Two vertices of an equilateral triangle are (–1, 0) and (1, 0) and its third vertex lies above the x-axis, the equation of the circumcircle is

1. 
2. 
3. 
4. 

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. The equation of the circle with centre $(–3,2)$ and radius $4\text{units}$ is

1. $x^2+y^2+6x-4y-7=0$
2. $x^2+y^2+6x-4y+3=0$
3. $x^2+y^2+6x-4y-13=0$
4. $x^2+y^2+6x-4y-3=0$

Q.

How do you write the equation of a circle given two points $?$

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 $\frac{\left|z-2\right|}{\left|z-3\right|}=2$ represents a circle, then its radius is equal to

1. $1$

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

3. $\frac{3}{4}$

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

Q.

The equation of one of the lines parallel to $4x-3y=5$ and at a unit distance from the point $(-1,-4)$ is

1. $3x+4y–3=0$

2. $3x+4y+3=0$

3. $4x–3y+3=0$

4. $4x–3y–3=0$

Q.

Suppose a circle passes through $\left(2,2\right)$ and $\left(9,9\right)$ and touches the $x-axisatP.$ If $O$ is the origin, then $OP$ is equal to

1. $4$

2. $5$

3. $6$

4. $9$

5. $11$

Q.

The point diametrically opposite to the point $P(1,0)$on the circle $x^2+y^2+2x+4y-3=0$ is

1. $(3,4)$

2. $(3,-4)$

3. $(-3,4)$

4. $(-3,-4)$

Q.

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

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.

If one end of the diameter is $(1,1)$ and other end lies on the line $x+y=3$, then locus of centre of circle is

1. $2(x-y)=5$
2. $2x+2y=5$
3. $x+y=1$
4. None of these

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

1. $x^2+y^2+2ay+2px-b^2-q^2=0$
2. $x^2+y^2+2ax+2py-b^2-q^2=0$
3. $x^2+y^2+2(a+p)x-b^2-q^2=0$
4. $x^2+y^2+2(a+p)y-b^2-q^2=0$

Q. If the equation $p{x}^2+(2-q)xy+3y^2-6qx+30y+6q=0$ represents a circle, then the value of $p\times q$ is

Q. The circle lying in the first quadrant whose centre lies on the curve $y=2x^2-27,$ has tangents as $4x-3y=0$ and the $y-$axis. Then the diameter of the circle (in units) is

Q. The equation of the circle inscribed in the triangle formed by the straight line $4x+3y=6$ and both the coordinate axes is

1. $4x^2+4y^2-4x-4y+1=0$
2. $2x^2+2y^2-4x-4y+1=0$
3. $5x^2+5y^2-5x-5y+11=0$
4. $x^2+y^2-6x-6y+18=0$

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

Q.

A track and field playing area is in the shape of a rectangle with semicircles at each end.

The inside perimeter of the track is to be $1500\mathrm{meters}$ .

What should the dimensions of the rectangle be so that the area of the rectangle is a maximum?

Q.

How do you write a polar equation for the circle centered at the origin with radius $\sqrt{2}$?

Q.

The smallest circle with center on $y$-axis and passing through the point $\left(7,3\right)$ has radius

1. $\sqrt{58}$

2. $7$

3. $3$

4. $4$

Q. A point moves in such a manner that the sum of the squares of its distances from the vertices of a triangle is constant. Then, the locus of the point is

1. a hyperbola
2. a parabola
3. an ellipse
4. a circle

Q. If the radius of the circle $x^2+y^2+2gx+2fy+c=0$ be r, then it will touch both the axes, if

1. 
2. 
3. g = f = r
4. g = f = c = r

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. If an equilateral triangle of side $a$ is inscribed in the circle $x^2+y^2-6x-4y+5=0,$ then the value of $a^2$ is

Q. The equation of the circle concentric with the circle $x^2+y^2-8x+6y-5=0$ and passing through the point $(-2,-7)$ is

1. $x^2+y^2-8x+6y-30=0$
2. $x^2+y^2-8x+6y-26=0$
3. $x^2+y^2-8x+6y-27=0$
4. $x^2+y^2-8x+6y=0$

Q.

The shortest distance between the circles ${\left(x-1\right)}^2+{\left(y+2\right)}^2=1$ and ${\left(x+2\right)}^2+{\left(y-2\right)}^2=4$ is

1. $1$

2. $2$

3. $3$

4. $4$

5. $5$

Q. The circle passing through the points $(1,0),(2,-7)$ and $(8,1)$ also passes through

1. $(9,6)$
2. $(9,-6)$
3. $(0,-3)$
4. $(0,3)$