U40 Locus and Equation of Circle - Essential Formulas

Standard Form

The equation of a circle with center at $(h, k)$ and radius $r$ is given by:

General Form

By expanding the standard form, we obtain the general form. The coefficients must satisfy the condition for it to represent a real circle ($g^2 + f^2 - c > 0$).

The center is $(-\frac{D}{2}, -\frac{E}{2})$ and the radius is $r = \sqrt{(\frac{D}{2})^2 + (\frac{E}{2})^2 - F}$.

Finding the Equation of a Locus

A locus is a set of points satisfying a given geometric condition. To find its equation:

1. Let $P(x, y)$ be a general point on the locus.
2. Translate the geometric condition into an algebraic equation involving $x$ and $y$.
3. Simplify the equation to its standard or general form.

Common conditions include: constant distance from a fixed point (circle), equal distances from two fixed points (perpendicular bisector), and constant ratio of distances from two fixed points (circle of Apollonius).

This represents the perpendicular bisector of the segment joining $(x_1, y_1)$ and $(x_2, y_2)$.

Number of Intersection Points

To find the intersection points, substitute the linear equation (e.g., $y = mx + c$) into the circle's equation. The resulting quadratic equation in $x$ (or $y$) determines the number of intersections based on its discriminant $\Delta$.

• If $\Delta > 0$, the line is a secant (two distinct intersection points).
• If $\Delta = 0$, the line is a tangent (one intersection point).
• If $\Delta < 0$, the line does not intersect the circle.

Equation of a Tangent

Given a point $P(x_1, y_1)$ on the circle $x^2 + y^2 + Dx + Ey + F = 0$, the equation of the tangent at $P$ is:

For the circle $(x - h)^2 + (y - k)^2 = r^2$, the tangent at $(x_1, y_1)$ is $(x_1 - h)(x - h) + (y_1 - k)(y - k) = r^2$.