This page presents a collection of math questions on circles and the distance between points. Complete answers are provided at the end of the page. These problems are suitable for high school and introductory precalculus students.
Find the distance between the points \( A(3,4) \) and \( B(5,8) \).
Find all values of \( x \) such that the distance between the points \( (x,4) \) and \( (-5,3) \) is equal to \( 5 \).
Write the equation of the circle with center at \( (0,0) \) and radius \( 6 \).
Find the center and radius of the circle whose equation is
\[ (x - 2)^2 + (y + 5)^2 = 13 \]Find the center and radius of the circle whose equation is
\[ (5 - x)^2 + (y - 1)^2 = 4 \]Find the center and radius of the circle whose equation is
\[ (-4 - x)^2 + (-y + 11)^2 = 9 \]Find the center and radius of the circle whose equation is
\[ x^2 + y^2 + 6x - 10y = 9 \]Find the center and radius of the circle whose equation is
\[ x^2 + y^2 + 4y = 0 \]Find the center and radius of the circle whose equation is
\[ -x^2 - y^2 + 8x = 0 \]Find the equation of the circle that has a diameter with endpoints \( (-6,1) \) and \( (2,-5) \).
Find the points of intersection of the circles
\[ x^2 + y^2 = 4 \] and \[ (x - 2)^2 + (y - 2)^2 = 4 \]Find the equation of the circle with center at \( (-3,5) \) that passes through the point \( (5,-1) \).
Find the equation of the circle that passes through the points \( (0,6) \), \( (0,0) \), and \( (8,0) \).
Find the points of intersection of the circle
\[ (x - 2)^2 + y^2 = 8 \]and the line
\[ y = x - 2 \]For which values of \( K \) is the line
\[ y = Kx \]tangent to the circle
\[ (x - 7)^2 + (y - 6)^2 = 9 \]For which value of \( K \) is the line
\[ x = 6 \]tangent to the circle
\[ x^2 + y^2 = K \]Determine whether the point \( (3,5) \) is inside, outside, or on the circle
\[ x^2 + y^2 = 9 \]For which values of \( K \) is the point \( (1,K) \) inside the circle
\[ x^2 + y^2 = 4 \]For which values of \( K \) is the point \( (K,-2) \) outside the circle
\[ x^2 + y^2 = 9 \]For which values of \( K \) is the point \( (K,2K) \) on the circle
\[ x^2 + y^2 = 5 \]1) \( 2\sqrt{5} \)
2) \( x = -7 \) or \( x = -3 \)
3) \[ x^2 + y^2 = 36 \]
4) Center \( (2,-5) \), radius \( \sqrt{13} \)
5) Center \( (5,1) \), radius \( 2 \)
6) Center \( (-4,11) \), radius \( 3 \)
7) Center \( (-3,5) \), radius \( \sqrt{43} \)
8) Center \( (0,-2) \), radius \( 2 \)
9) Center \( (4,0) \), radius \( 4 \)
10) \[ (x + 2)^2 + (y + 2)^2 = 25 \]
11) \( (2,0) \) and \( (0,2) \)
12) \[ (x + 3)^2 + (y - 5)^2 = 100 \]
13) \[ (x - 4)^2 + (y - 3)^2 = 25 \]
14) \( (0,-2) \) and \( (4,2) \)
15) \[ K = \frac{21 \pm 3\sqrt{19}}{20} \] Approximations: \( K \approx 1.7 \) and \( K \approx 0.4 \)
16) \( K = 36 \)
17) Outside
18) \( -\sqrt{3} < K < \sqrt{3} \)
19) \[ K \in (-\infty,-\sqrt{5}) \cup (\sqrt{5},\infty) \]
20) \( K = \pm 1 \)