Equation of Circle with (h,k) as Center

Q. The circle passing through $(1,-2)$ and touching the axis of $x$ at $(3,0)$ also passes through the point

1. $(-5,2)$
2. $(2,-5)$
3. $(-2,5)$
4. $(5,-2)$

Q.

The number of integral values of $k$ for which the line, $3x+4y=k$ intersects the circle, $x^2+y^2-2x-4y+4=0$ at two distinct points is

Q.

The circle passing through the intersection of the circles ${\mathrm{x}}^2+{\mathrm{y}}^2-6\mathrm{x}=0$ and, $x^2+y^2-4y=0$, having its center on the line, $2\mathrm{x}-3\mathrm{y}+12=0,$ also passes through the point:

1. $(-1,3)$

2. $(1,-3)$

3. $(-3,6)$

4. $(-3,1)$

Q. A circle touches the $y$-axis at the point $(0,4)$ and passes through the point $(2,0)$. Which of the following lines is not a tangent to the circle?

1. $4x+3y-8=0$
2. $3x-4y-24=0$
3. $3x+4y-6=0$
4. $4x-3y+17=0$

Q.

Find the equation of the circle which has its centre at the point (3, 4) and touches the straight line $5x+12y-1=0$

Q.

The equation of the circle which touches x the axes of coordinates and the line $\frac{x}{3}+\frac{y}{4}=1$ and whose centres lie in the first quadrant is $x^2+y^2-2cx-2cy+c^2=0$ where c is equal to

1. 4

2. 2

3. 3

4. 6

Q. Let differential equation of family of circles touching $y-$axis at the origin be $\frac{dy}{dx}+\frac{x^2-\lambda y^2}{\mu xy}=0$, where $\lambda \in R$, $\mu \in R-\left\{0\right\}$. Then the value of $\lambda +\mu$ is

Q.

The equation of the smallest circle passing through the intersection of the line $x+y=1$ and the circle $x^2+y^2=9$ is

1. $x^2+y^2+x+y-8=0$

2. $x^2+y^2-x-y-8=0$

3. $x^2+y^2-x+y-8=0$

4. None of these

Q.

Find the centre and radius of the circle x² + y² – 4x – 8y – 45 = 0

Q.

The line segment joining points$(4,7)$and $(-2,-1)$is the diameter of a circle. If the circle intersects the $X-$axis at$A$ and$B$, then$AB$ is equal to

1. $4$

2. $5$

3. $6$

4. $8$

Q.

Find the equation of a circle passing through the point $(7,3)$ having radius $3$ units and whose centre lies on the line $y=x-1$.

Q. P is a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, N is the foot of the perpendicular from P on the transverse axis.The tangent to the hyperbola at P meets the transverse axis at T. If O is the centre of the hyperbola, then OT.ON is equal to

1. 
2. 
3. 
4. 

Q. The locus of the centres of the circles, which touch the circle, $x^2+y^2=1$ externally, also touch the $y-$axis and lie in the first quadrant, is:

1. $y=\sqrt{1+2x},x\ge 0$
2. $y=\sqrt{1+4x},x\ge 0$
3. $x=\sqrt{1+4y},y\ge 0$
4. $x=\sqrt{1+2y},y\ge 0$

Q.

Equation of the circle through origin which cuts intercepts of length a and b on axes is

1. $x^2+y^2-ax-by=0$
   

2. none of these

3. $x^2+y^2+ax+by=0$

4. $x^2+y^2+bx+ay=0$

Q.

A circle has radius 3 units and its centre lies on the line y = x - 1. If it passes through the point (7, 3), find its equations.

Q.

Find the equation of the circle with centre (0, 2) and radius 2

Q.

The focal distance of a point on the parabola $y^2=16x$, whose ordinate is twice the abscissa, is

1. $6$

2. $8$

3. $10$

4. $12$

Q.

The point $(-4,5)$ is the vertex of a square and one of its diagonals is $7x-y+8=0$. The equation of the other diagonal is

1. $7x–y+23=0$

2. $7y+x=30$

3. $7y+x=31$

4. $x–7y=30$

Q.

The equations of the circle which pass through the origin and make intercepts of lengths $4$ and $8$ on the $x$ and $y$ - axes respectively are

1. $x^2+y^2\pm 4x\pm 8y=0$

2. $x^2+y^2\pm 2x\pm 4y=0$

3. $x^2+y^2\pm 8x\pm 16y=0$

4. $x^2+y^2\pm x\pm y=0$

Q.

If $z=\left(\lambda +3\right)+i\sqrt{5-{\lambda }^2}$, then the locus of $z$ is a

1. circle

2. straight line

3. parabola

4. None of these

Q.

One diameter of the circle circumscribing the rectangle ABCD is 4y= x + 7.
If the coordinates of A and B are (-3, 4) and (5, 4) respectively, find the equation of the circle.

Q. The equation of the circle whose diameter lies on 2x + 3y = 3 and 16x - y = 4 which passes through (4, 6) is

1. $x^2+y^2-4x-8y=200$
2. $5(x^2+y^2)-4x=200$
3. $x^2+y^2=40$
4. $5(x^2+y^2)-3x-8y=200$

Q. If two adjacent sides of a cyclic quadrilateral are $2$ and $5$ and the angle between them is ${60}^{\circ }$. If the third side is $3$, then the remaining fourth side is

1. $2$
2. $3$
3. $4$
4. $5$

Q.

Choose the correct statement about two circles whose equations are given below:

$x^2+y^2–10x–10y+41=0$

$x^2+y^2–22x–10y+137=0$

1. Circles have no meeting point.

2. Circles have two. meeting point

3. Circles have only one meeting point.

4. Circles have the same center.

Q. The equation of circle whose centre lies on 3x - y - 4 = 0 and x + 3y + 2 = 0 and has an area 154 square units is

1. $x^2+y^2-2x+2y-47=0$
2. $x^2+y^2-2x+2y+47=0$
3. None of these
4. $x^2+y^2+2x-2y-47=0$

Q.

Find the equation of a circle whose centre is (3, -1) and which cuts off a chord of length 6 units on the line $2x-5y+18=0$.

Q.

Find the equations of the circles passing through two points on y-axis at distance $3$ from the origin and having radius $5$.

Q. The centre of a circle is (2, –3) and the circumference is $10\pi .$ Then the equation of the circle is

1. $x^2+y^2-4x+6y+12=0$
2. $x^2+y^2+4x+6y+12=0$
3. $x^2+y^2-4x+6y-12=0$
4. $x^2+y^2-4x-6y-12=0$

Q. If two adjacent vertices of a regular hexagon are $(0,0)$ and $(1,2),$ then equation of the circumcircle of the hexagon is

1. $x^2+y^2-x-2y=\pm 3$
2. $x^2+y^2-x-2y=\pm \sqrt{3}$
3. $x^2+y^2-x-2y=\pm \sqrt{3}(2x-y)$
4. $x^2+y^2-x-2y=0$

Q. If a circle passes through the point $A(1,0),$ $B(5,0)$ and touches $y-$axis at $C(0,h),$ then the value of $h^2$ is