Circles Flashcards
What is the general form of the equation of a circle?
The general form is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
Given the equation of a circle (x + 3)^2 + (y - 4)^2 = 25, find the center and radius.
The center is (-3, 4) and the radius is 5.
How can you rewrite the equation x^2 + y^2 + 6x - 8y = 0 in the standard form of a circle?
Complete the square for the x and y terms to get (x + 3)^2 + (y - 4)^2 = 25.
How do you find the equation of a circle passing through three points, say A(1, 2), B(3, 4), and C(5, 6)?
Use the general form of the circle equation and substitute the coordinates of the points to create a system of equations.
What is the condition for a line y = mx + c to be a tangent to the circle x^2 + y^2 = r^2?
The perpendicular distance from the center of the circle to the line must be equal to the radius r.
What is the equation of a tangent line to the circle x^2 + y^2 = r^2?
A tangent line to the circle x^2 + y^2 = r^2 has the form xcosθ + ysinθ = r, where θ is the angle the radius makes with the x-axis.
How do you find the length of the tangent from a point (x1, y1) to a circle (x - h)^2 + (y - k)^2 = r^2?
The length of the tangent is √((x1 - h)^2 + (y1 - k)^2 - r^2).
What is the formula for the equation of a circle with center (h, k) and radius r?
The formula is (x - h)^2 + (y - k)^2 = r^2.
How can you determine if a given point lies on the circle x^2 + y^2 = r^2?
Substitute the coordinates of the point into the equation. If it satisfies the equation, the point lies on the circle.
What is the equation of a circle that has its center at the origin and a radius of 3?
The equation is x^2 + y^2 = 9.
How do you find the coordinates of the center and the radius of a circle given its equation in general form Ax^2 + By^2 + Cx + Dy + E = 0?
Complete the square for both x and y terms to convert it to the standard form (x - h)^2 + (y - k)^2 = r^2.
What is the radius of the circle with the equation (x - 4)^2 + (y + 2)^2 = 25?
The radius is 5 because √25 = 5.
How do you find the center of the circle with the equation x^2 + y^2 + 6x - 8y = -4?
Complete the square for the x and y terms to find the center (-3, 4).
How do you find the equation of a circle passing through two points A(x1, y1) and B(x2, y2)?
Find the midpoint of A and B to get the center, and calculate the radius as the distance from the center to either point.
What is the formula for the distance between two points on a circle that are diametrically opposite?
The distance is twice the radius of the circle, 2r.
What is the equation of a circle with center (3, -2) and radius 4?
The equation is (x - 3)^2 + (y + 2)^2 = 16.
How do you find the equation of a circle with a given center and passing through a given point?
Use the center to find the equation (x - h)^2 + (y - k)^2, then use the given point to solve for the radius.
What is the equation of a circle with center (0, 0) and radius r?
The equation is x^2 + y^2 = r^2.
How do you find the center and radius of a circle given in the form x^2 + y^2 + 10x - 4y = 0?
Complete the square for both x and y terms to convert it to the standard form.
What is the relationship between the radius of a circle and the length of a chord?
The radius of the circle is always longer than the length of any chord that does not pass through the center.
