Topic 1: Standard & Simplest Equations

1. Find the equation of a circle with center $(0,0)$ and radius $5$ units.
2. Find the equation of a circle with center $(2, -3)$ and radius $4$.
3. Find the center and radius of the circle given by $(x - 5)^2 + (y + 2)^2 = 36$.
4. Write the equation of a circle passing through the origin with radius $3$ and center lying on the positive x-axis.
5. Find the equation of a circle whose center is $(-1, 2)$ and which passes through the point $(3, 5)$.
6. A circle has its center at $(3, 4)$ and touches the y-axis. Find its equation.
7. Find the equation of a circle with radius $\sqrt{5}$ and center at the origin.
8. Does the point $(3, 4)$ lie on the circle $x^2 + y^2 = 25$?

Topic 2: General Equation (Center & Radius)

1. Find the center and radius of the circle $x^2 + y^2 - 4x - 8y - 45 = 0$.
2. Find the center and radius of the circle $x^2 + y^2 + 8x + 10y - 8 = 0$.
3. Find the center and radius of the circle $2x^2 + 2y^2 - x = 0$.
4. Determine the equation of the circle passing through points $(1,0)$, $(-1,0)$, and $(0,1)$.
5. Find the equation of the circle concentric with $x^2 + y^2 - 6x + 12y + 15 = 0$ and having double its area.
6. Show that the equation $x^2 + y^2 + 4x - 6y + 13 = 0$ represents a point circle.
7. Find the value of $k$ if the equation $kx^2 + y^2 - 4x + 6y - 3 = 0$ represents a circle.
8. If the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ passes through the origin, find the value of $c$.
9. Find the equation of the circle passing through three points $(0,0)$, $(a,0)$, and $(0,b)$.
10. Find the center of the circle $3x^2 + 3y^2 - 12x + 15y - 23 = 0$.

Topic 3: Nature of Circle

1. Check if $x^2 + y^2 - 2x + 4y + 10 = 0$ represents a real circle.
2. Determine the nature of the circle represented by $x^2 + y^2 + 2x - 6y + 10 = 0$.
3. For what values of $c$ does the equation $x^2 + y^2 - 4x - 2y + c = 0$ represent a real circle?
4. Prove that $x^2 + y^2 + 2x + 2y + 5 = 0$ represents an imaginary circle.
5. Find the radius of the circle $x^2 + y^2 - 4x - 4y + 8 = 0$. What does this radius imply?

Topic 4: Diameter Form

1. Find the equation of the circle drawn on the line segment joining $(1, 2)$ and $(3, -4)$ as diameter.
2. Find the equation of the circle with diameter endpoints $(-2, 3)$ and $(-3, 5)$.
3. Find the equation of the circle if the coordinates of the diameter are origin and $(4, 4)$.
4. The line $4x + 3y - 24 = 0$ intersects the axes at A and B. Find the equation of the circle described on AB as diameter.
5. If one end of a diameter of the circle $x^2 + y^2 - 4x - 6y + 11 = 0$ is $(3, 4)$, find the coordinates of the other end.
6. Find the equation of the circle having $(0, 0)$ and $(2, 2)$ as ends of a diameter.
7. A rectangle ABCD has vertices A$(1,2)$, B$(1,6)$, C$(5,6)$, and D$(5,2)$. Find the equation of the circumcircle of this rectangle. (Hint: Diagonal AC is a diameter).

Topic 5: Parametric Equations

1. Write the parametric equations of the circle $x^2 + y^2 = 16$.
2. Find the parametric equations of the circle $(x - 1)^2 + (y + 2)^2 = 25$.
3. If the parametric equations are $x = 3 + 2\cos\theta$ and $y = 4 + 2\sin\theta$, find the Cartesian equation.
4. Find the center and radius of the circle given by $x = -1 + 3\cos\theta, y = 2 + 3\sin\theta$.
5. For the circle $x^2 + y^2 + 4x - 6y - 12 = 0$, find its parametric representation.

Topic 6: Position of a Point

1. Determine the position of point $(2, 1)$ with respect to the circle $x^2 + y^2 = 9$.
2. Does the point $(1, -2)$ lie inside, outside, or on the circle $x^2 + y^2 - 4x + 2y - 11 = 0$?
3. Find the value of $k$ if the point $(1, 2)$ lies on the circle $x^2 + y^2 - 2x + 6y + k = 0$.
4. Find the position of the point $(-3, -4)$ with respect to the circle $x^2 + y^2 = 20$.
5. Find the range of values of $a$ if the point $(a, a)$ lies inside the circle $x^2 + y^2 = 8$.
6. Check if the center of the circle $x^2 + y^2 - 6x - 8y + 5 = 0$ lies inside the circle $x^2 + y^2 = 1$.
7. For the circle $x^2 + y^2 - 4x + 6y - 3 = 0$, find the power of point $(1, 1)$.

Topic 7: Intercepts & Line Interactions

1. Find the length of the intercept made by the circle $x^2 + y^2 + 8x - 4y - 5 = 0$ on the x-axis.
2. Find the length of the y-intercept for the circle $x^2 + y^2 - 10x + 6y + 4 = 0$.
3. Does the circle $x^2 + y^2 + 2x + 2y + 5 = 0$ cut the x-axis?
4. Find the condition for the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ to touch the x-axis.
5. Find the length of the chord cut off by the circle $x^2 + y^2 = 25$ on the line $y = 3$.
6. Determine if the line $3x + 4y - 25 = 0$ is a tangent, secant, or exterior to the circle $x^2 + y^2 = 25$.
7. Find the length of the x-intercept of the circle passing through origin and having center at $(3, 4)$.
8. Find the length of the intercept on the y-axis made by the circle $x^2 + y^2 - 5x - 13y + 30 = 0$.