Circle standard and general equations and properties - One Line Questions
1.
The equation x^2 + y^2 + 2x - 4y + k = 0 represents a circle. If the radius is 3, find the value of k. —
-6
2.
What is the center of the circle represented by Ax^2 + Ay^2 + 2Gx + 2Fy + C = 0? —
(-G/A, -F/A)
3.
What is the center of the circle given by the equation x^2 + y^2 - 4x + 6y - 12 = 0? —
(2, -3)
4.
The equation 2x^2 + 2y^2 - 3x + 5y - 7 = 0 represents a circle. What is its center? —
(3/4, -5/4)
5.
What is the center of the circle passing through the points (0,0), (a,0), and (0,b)? —
(a/2, b/2)
6.
For the general equation of a circle x^2 + y^2 + 2gx + 2fy + c = 0, what are the coordinates of the center? —
(-g, -f)
7.
The equation of a circle is x^2 + y^2 = r^2. What is its center? —
(0, 0)
8.
Find the equation of the circle passing through (1, 2) and having its center at the intersection of 2x + y = 3 and x - y = 0. —
(x - 1)^2 + (y - 1)^2 = 2
9.
What is the equation of the circle with center (1, 2) passing through the point (4, 6)? —
(x - 1)^2 + (y - 2)^2 = 25
10.
What is the equation of the circle whose center lies on the line y = x and passes through (2, 0) and (0, 4)? —
(x - 2)^2 + (y - 2)^2 = 8
11.
Find the equation of the circle with center (2, -3) and radius 4. —
(x - 2)^2 + (y + 3)^2 = 16
12.
A circle passes through the origin (0,0) and has its center at (3,4). What is its equation? —
(x - 3)^2 + (y - 4)^2 = 25
13.
Find the equation of the circle with center (3, -2) and touching the line 3x - 4y + 5 = 0. —
(x - 3)^2 + (y + 2)^2 = (26/5)^2
14.
Find the equation of the circle which touches both axes and has its center in the first quadrant and radius 5. —
(x - 5)^2 + (y - 5)^2 = 25
15.
What is the equation of the circle whose center is (5, -4) and circumference is 12π? —
(x - 5)^2 + (y + 4)^2 = 36
16.
Find the equation of the circle whose diameter has endpoints (1, 2) and (4, 5). —
(x - 5/2)^2 + (y - 7/2)^2 = 25/4
17.
What is the equation of the circle with center (a cos θ, a sin θ) and radius a? —
(x - a cos θ)^2 + (y - a sin θ)^2 = a^2
18.
What is the equation of the circle passing through (a, b) and concentric with x^2 + y^2 + 2gx + 2fy + c = 0? —
(x - g)^2 + (y - f)^2 = (a - g)^2 + (b - f)^2
19.
Find the equation of the circle with center (h, k) touching the y-axis. —
(x - h)^2 + (y - k)^2 = h^2
20.
Find the equation of the circle with center (h, k) touching the x-axis. —
(x - h)^2 + (y - k)^2 = k^2
21.
What is the standard equation of a circle with center (h, k) and radius r? —
(x - h)^2 + (y - k)^2 = r^2
22.
The equation (x-a)^2 + (y-b)^2 = 0 represents: —
A circle with center (a,b) and radius 0
23.
If g^2 + f^2 - c = 0 for the general equation of a circle, what does the equation represent? —
A point circle
24.
If g^2 + f^2 - c < 0 for the general equation of a circle, what does the equation represent? —
An imaginary circle
25.
The equation x^2 + y^2 = 0 represents: —
A point circle at the origin
26.
What is the locus of a point that moves such that its distance from a fixed point is constant? —
A circle
27.
The equation x^2 + y^2 + 2gx + 2fy + c = 0 represents a circle. What is the condition for it to pass through the origin? —
c = 0
28.
What is the condition for the line y = mx + c to be a tangent to the circle x^2 + y^2 = a^2? —
c^2 = a^2(m^2 + 1)
29.
The equation x^2 + y^2 = a^2 is the standard equation of a circle with: —
Center (0, 0) and radius a
30.
The equation Ax^2 + Ay^2 + 2Gx + 2Fy + C = 0 represents a circle if: —
A(G^2 + F^2) - C > 0
31.
The equation x^2 + y^2 + 2gx + 2fy + c = 0 represents a circle of infinite radius when: —
A = 0 (in Ax^2 + Ay^2...)
32.
The equation x^2 + y^2 + 2gx + 2fy + c = 0 represents a circle if: —
g^2 + f^2 - c > 0
33.
If the two circles x^2 + y^2 + 2g1x + 2f1y + c1 = 0 and x^2 + y^2 + 2g2x + 2f2y + c2 = 0 are concentric, then: —
g1 = g2, f1 = f2
34.
If the power of a point P with respect to a circle is zero, then P lies: —
On the circle
35.
If the power of a point P with respect to a circle is negative, then P lies: —
Inside the circle
36.
If the power of a point P with respect to a circle is positive, then P lies: —
Outside the circle
37.
The equation x^2 + y^2 = r^2 represents a circle with radius: —
r
38.
What is the radius of the circle given by the equation x^2 + y^2 - 4x + 6y - 12 = 0? —
sqrt(25)
39.
The equation 2x^2 + 2y^2 - 3x + 5y - 7 = 0 represents a circle. What is its radius? —
sqrt(9 + 25 + 28)/4
40.
What is the radius of the circle passing through the points (0,0), (a,0), and (0,b)? —
sqrt(a^2 + b^2)/2
41.
For the general equation of a circle x^2 + y^2 + 2gx + 2fy + c = 0, what is the radius? —
sqrt(g^2 + f^2 - c)
42.
What is the radius of the circle represented by Ax^2 + Ay^2 + 2Gx + 2Fy + C = 0? —
sqrt(G^2/A^2 + F^2/A^2 - C/A)
43.
What is the length of the tangent from an external point (x1, y1) to the circle x^2 + y^2 + 2gx + 2fy + c = 0? —
sqrt(x1^2 + y1^2 + 2gx1 + 2fy1 + c)
44.
What is the equation of the circle whose center is at the origin and radius is R? —
x^2 + y^2 = R^2
45.
What is the equation of the circle passing through (0,0) and concentric with x^2 + y^2 - 4x + 6y - 12 = 0? —
x^2 + y^2 = 13
46.
What is the equation of the circle passing through the origin and cutting the axes at intercepts a and b on x-axis and y-axis respectively? —
x^2 + y^2 - ax - by = 0
47.
Find the equation of the circle passing through the points (0,0), (a,0), and (0,b). —
x^2 + y^2 - ax - by = 0
48.
What is the general equation of a circle? —
x^2 + y^2 + 2gx + 2fy + c = 0
49.
Find the equation of the circle passing through the points (1, 1), (1, -1), and (-1, 1). —
x^2 + y^2 = 1
50.
What is the power of the point (x1, y1) with respect to the circle S = x^2 + y^2 + 2gx + 2fy + c = 0? —
x1^2 + y1^2 + 2gx1 + 2fy1 + c