Gr12 Mathematics: Ch 6.1 Equation of a Circle

What is the equation of a circle with center at the origin and radius r?

x^2 + y^2 = r

x - y = r

x + y = r

x^2 + y^2 = r^2 (correct)

A circle with center at the origin is symmetric about:

The lines y = x and y = -x only

The x-axis and the y-axis only

The origin only

The x-axis, the y-axis, the origin, and the lines y = x and y = -x (correct)

What is the equation of a circle with center at (a, b) and radius r?

(x - a)^2 + (y - b)^2 = r^2 (correct)

(x - a) + (y - b) = r

(x - a) - (y - b) = r

(x - a)^2 - (y - b)^2 = r^2

What is the purpose of completing the square in a circle's equation?

To rewrite the equation in standard form Signup and view all the answers

What is the first step in completing the square to find the center and radius of a circle?

Group the x terms and the y terms Signup and view all the answers

What is the equation derived from when deriving the equation of a circle with center at the origin?

The Pythagorean theorem Signup and view all the answers

What is the formula used to find the distance from a point (x, y) to the origin?

OP = sqrt((x - 0)^2 + (y - 0)^2) Signup and view all the answers

What is the general form of a circle's equation that can be rewritten in standard form by completing the square?

x^2 + y^2 + Dx + Ey + F = 0 Signup and view all the answers

What is the radius of a circle with the equation x^2 + y^2 = 16?

4 Signup and view all the answers

If a circle has the equation (x - 2)^2 + (y - 3)^2 = 9, what are the coordinates of its center?

(2, 3) Signup and view all the answers

What is the graphical representation of the equation x^2 + y^2 = 0?

A point at the origin Signup and view all the answers

If a circle has the equation x^2 + y^2 + 4x - 6y - 3 = 0, what is the next step in completing the square?

Group the x terms and the y terms Signup and view all the answers

What is the benefit of completing the square in a circle's equation?

To find the equation of the circle in standard form Signup and view all the answers

If a circle has the equation (x + 1)^2 + (y - 2)^2 = 4, which of the following is NOT true about the circle?

The circle is symmetric about the y-axis Signup and view all the answers

What is the purpose of using the distance formula when deriving the equation of a circle?

To find the distance from the center to a point on the circle Signup and view all the answers

If a circle has the equation x^2 + y^2 - 6x + 8y - 11 = 0, what can be said about the circle?

The circle has a radius that can be determined by completing the square Signup and view all the answers

If the equation of a circle is x^2 + y^2 - 4x - 8y + 5 = 0, what is the radius of the circle?

.sqrt(16) Signup and view all the answers

What is the distance between the centers of two circles with equations (x - 1)^2 + (y - 2)^2 = 9 and (x + 2)^2 + (y + 1)^2 = 4?

sqrt(65) Signup and view all the answers

If a circle has the equation x^2 + y^2 - 10x + 2y + 25 = 0, what is the x-coordinate of its center?

-5 Signup and view all the answers

If two circles have equations (x + 1)^2 + (y - 2)^2 = 4 and (x - 3)^2 + (y + 1)^2 = 9, what can be said about the circles?

They do not intersect Signup and view all the answers

If a circle has the equation x^2 + y^2 - 6x - 8y + 12 = 0, what is the equation of its circumcircle?

x^2 + y^2 - 6x - 8y + 12 = 0 Signup and view all the answers

If a circle has the equation x^2 + y^2 - 4x - 2y + 1 = 0, what is the y-coordinate of its center?

1/2 Signup and view all the answers

If two circles have equations x^2 + y^2 - 6x + 2y + 3 = 0 and x^2 + y^2 + 2x - 4y - 1 = 0, what can be said about the circles?

They intersect at two points Signup and view all the answers

If a circle has the equation x^2 + y^2 - 8x + 2y + 7 = 0, what is the radius of the circle?

sqrt(10) Signup and view all the answers