# Mathematics (I)
# Unit 4: Straight Lines and Circles
## 4-3 Relationship Between Circles and Lines
### Topic 1: Definition and Equation of a Circle

(1) Definition of a Circle:
- In a plane, the set of all points that are equidistant from a fixed point forms a circle.
- The fixed point is called the center of the circle, and the distance from the center to any point on the circle is called the radius, as shown in the figure.

(2) Standard Form of a Circle:
- If the center is $A(h, k)$ and the radius is $r$, then the equation of the circle is
$$
(x-h)^2 + (y-k)^2 = r^2
$$

(3) General Form of a Circle:
- Expanding the standard form gives a quadratic equation in two variables, known as the general form of a circle:
$$x^2 + y^2 + dx + ey + f = 0$$

- Completing the square with respect to $x$ and $y$ gives:
$$\left( x + \frac{d}{2} \right)^2 + \left( y + \frac{e}{2} \right)^2 = \frac{1}{4} \left( d^2 + e^2 - 4f \right)$$

(4) Graph of the Quadratic Equation $x^2 + y^2 + dx + ey + f = 0$:

* If $d^2 + e^2 - 4f > 0$, the graph represents a circle with
center $\left( -\frac{d}{2}, -\frac{e}{2} \right)$ and radius $\frac{1}{2} \sqrt{d^2 + e^2 - 4f}$.

- If $d^2 + e^2 - 4f = 0$, the graph represents a single point at
$\left( -\frac{d}{2}, -\frac{e}{2} \right)$.

- If $d^2 + e^2 - 4f < 0$, the equation has no real graph (no circle exists).

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### Topic 2: Relationship Between a Circle and a Point

Given a circle $C: x^2 + y^2 + dx + ey + f = 0$ with center $A$ and radius $r$, and a point $P(x_0, y_0)$:

(1) Point $P$ is inside the circle:
When $\overline{PA} < r$, as shown in Figure (1), $$ x_0^2 + y_0^2 + dx_0 + ey_0 + f < 0$$

(2) Point $P$ is on the circle:
When $\overline{PA} = r$, as shown in Figure (2), $$ x_0^2 + y_0^2 + dx_0 + ey_0 + f = 0$$

(3) Point $P$ is outside the circle:
When $\overline{PA} > r$, as shown in Figure (3),$$ x_0^2 + y_0^2 + dx_0 + ey_0 + f > 0$$


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### Topic 3: The Relationship Between a Circle and a Line

(1) The relationship between a circle and a straight line on a plane can be categorized into the following three cases:

* **Intersects at Two Points (Secant):** The circle and the line intersect at two distinct points. In this case, the line is called a **secant line** of the circle, as shown in Figure (I).

* **Tangent (Intersects at One Point):** The circle and the line intersect at exactly one point. In this case, the line is called a **tangent line** of the circle, and the intersection point is called the **point of tangency**, as shown in Figure (II).

* **Do Not Intersect:** The circle and the line have no intersection points, as shown in Figure (III).


(2) Geometric Discrimination Method (Distance Method):

Let the center of the circle $C$ be $A$, the radius be $r$, and the distance from the center to the line $L$ be $d$. The relationship between $d$ and $r$ can be categorized into the following three cases:

* When $d < r$, the circle $C$ and the line $L$ **intersect at two points**, as shown in Figure (IV).

* When $d = r$, the circle $C$ and the line $L$ **intersect at one point** (tangent), as shown in Figure (V).

* When $d > r$, the circle $C$ and the line $L$ **do not intersect**, as shown in Figure (VI).

(3) Algebraic Discrimination Method (Discriminant Method):

Let the circle be $C: x^2 + y^2 + dx + ey + f = 0$ and the line be $L: ax + by + c = 0$. If $b \neq 0$, rewrite $L$ as $y= -\frac{a}{b}x - \frac{c}{b}$

Substituting this expression for $y$ into the equation for circle $C$ yields a quadratic equation in $x$: $Ax^2 + Bx + C = 0$. By using the **discriminant** $D = B^2 - 4AC$, the following three cases can be determined:

* When $D > 0$, the quadratic equation has two distinct real roots, which means the circle $C$ and the line $L$ **intersect at two distinct points**.
* When $D = 0$, the quadratic equation has two equal real roots, which means the circle $C$ and the line $L$ **intersect at one point** (tangent).
* When $D < 0$, the quadratic equation has no real roots, which means the circle $C$ and the line $L$ **do not intersect**.

> Note: When $b = 0$, the line equation is $x = -\frac{c}{a}$. Substituting this into the circle equation $C$ gives a quadratic equation in $y$, and the discussion process is the same as above.

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### Topic 4: Tangent Lines of a Circle

(1) Let $P(x_0, y_0)$ be a fixed point on the circle $C: (x-h)^2 + (y-k)^2 = r^2$, and let line $L$ be the tangent line passing through point P:

* The slope of the tangent line $L$ is $-\frac{x_0-h}{y_0-k}$. This is because the slope of the line segment $AP$ (where $A$ is the center $(h, k)$) is $\frac{y_0-k}{x_0-h}$, and $AP$ is perpendicular to $L$.
* The equation of the tangent line $L$ is $$y-y_0=-\frac{x_0-h}{y_0-k}(x-x_0)$$

(2) Let $P(x_0, y_0)$ be a fixed point outside the circle $C: (x-h)^2 + (y-k)^2 = r^2$:

* There are two tangent lines passing through point $P$.
* Let the equation of the tangent line passing through $P$ be $y-y_0=m(x-x_0)$. The value(s) of $m$ can be found by using the condition that the distance from the circle's center to the tangent line must equal the radius.
* If only one solution for $m$ is found, it indicates that the other tangent line is the vertical line $x-x_0=0$.

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