What is the geometric definition of a circle as a locus of points?
A circle is the set of all points in a plane that are at a fixed distance (the radius, \(r\)) from a fixed coplanar point (the center, \((h, k)\)).
How is the standard equation of a circle derived using the distance formula?
Set the Euclidean distance between any point \((x, y)\) on the circle and the center \((h, k)\) equal to \(r\):
$$\sqrt{(x - h)^2 + (y - k)^2} = r$$
Squaring both sides yields:
$$(x - h)^2 + (y - k)^2 = r^2$$
What is the standard form equation of a circle with center \((h, k)\) and radius \(r\)?
$$(x - h)^2 + (y - k)^2 = r^2$$
What is the standard equation of a circle centered at the origin \((0, 0)\) with radius \(r\)?
$$x^2 + y^2 = r^2$$
What is the general form of the equation of a circle in Cartesian coordinates?
$$x^2 + y^2 + Dx + Ey + F = 0$$
Where \(D\), \(E\), and \(F\) are real constants, and the coefficients of \(x^2\) and \(y^2\) are equal and non-zero.
How do you convert the general equation \(x^2 + y^2 + Dx + Ey + F = 0\) into standard form?
Complete the square for the \(x\) and \(y\) terms separately:
$$\left(x + \frac{D}{2}\right)^2 + \left(y + \frac{E}{2}\right)^2 = \frac{D^2 + E^2 - 4F}{4}$$
In terms of the general equation coefficients \(D\), \(E\), and \(F\), what are the coordinates of the circle's center?
$$\left(-\frac{D}{2}, -\frac{E}{2}\right)$$
In the general form of a circle, what does the expression \(D^2 + E^2 - 4F\) indicate?
1. If \(D^2 + E^2 - 4F > 0\): A real circle with \(r = \frac{1}{2}\sqrt{D^2 + E^2 - 4F}\).
2. If \(D^2 + E^2 - 4F = 0\): A point circle at the center.
3. If \(D^2 + E^2 - 4F < 0\): No real locus (imaginary circle).
How do you determine the center and radius of a circle given the endpoints of its diameter, \((x_1, y_1)\) and \((x_2, y_2)\)?
1. Center \((h, k)\) is the midpoint: \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\).
2. Radius \(r\) is half the distance: \(r = \frac{1}{2}\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
What geometric property defines the line tangent to a circle at a point \(P(x_1, y_1)\)?
The tangent line is perpendicular to the radial segment connecting the center \((h, k)\) to the point of tangency \(P(x_1, y_1)\).
What are the center and radius of the circle defined by \((x - 4)^2 + (y + 7)^2 = 81\)?
The center is \((4, -7)\) and the radius is \(r = \sqrt{81} = 9\).
Convert \(x^2 + y^2 - 6x + 8y = 0\) into standard form.
Group and complete the square:
$$(x^2 - 6x + 9) + (y^2 + 8y + 16) = 9 + 16$$
$$(x - 3)^2 + (y + 4)^2 = 25$$
Center is \((3, -4)\), radius is \(5\).
Why must the coefficients of \(x^2\) and \(y^2\) be identical for a quadratic equation to represent a circle?
Equal coefficients ensure the geometric scaling along both the \(x\)-axis and \(y\)-axis is identical, producing uniform circular curvature rather than an ellipse.
What is the equation of the unit circle centered at the origin?
