Position of a Line to a Circle

Understanding the Relationship Between Lines and Circles

Imagine you have a circle and a straight line on the same plane. Interesting, right? How can these two geometric objects interact? It turns out there are three possibilities that can occur.

The line can intersect the circle at two points, be tangent to the circle at one point only, or even not touch the circle at all. Just like when you throw a pencil toward a ring, the pencil can go through the ring, touch the edge of the ring, or miss completely.

This concept is very important in analytic geometry because it helps us understand various situations in real life. For example, to determine whether a highway will pass through a circular protected area, or to analyze satellite trajectories relative to certain zones.

Three Possible Positions

Let's look at the three possible positions of a line relative to a circle visually:

From the visualization above, we can see three different situations. The first line doesn't touch the circle at all, the second line touches the circle at exactly one point, and the third line penetrates the circle so it intersects at two points.

1. Line intersects the circle occurs when a straight line passes through the interior of the circle so that it meets the circumference of the circle at two different points.

2. Line is tangent to the circle occurs when a straight line only touches the circumference of the circle at exactly one point. Such a line is called a tangent line.

3. Line does not intersect occurs when a straight line is outside the circle so there is no point of intersection between the line and the circle.

Discriminant Method

To determine the position of a line relative to a circle mathematically, we use the substitution method which produces a quadratic equation. Then, we analyze the discriminant of that quadratic equation.

Suppose we have a line with equation $y = mx + c$ and a circle with equation $x^2 + y^2 + Dx + Ey + F = 0$.

The first step is to substitute the line equation into the circle equation. It's easy, just replace all $y$ in the circle equation with $mx + c$:

This substitution result forms a quadratic equation in the form $ax^2 + bx + c = 0$ with coefficients:

Now, we calculate the discriminant of this quadratic equation. The discriminant is a value that determines the type of roots of a quadratic equation:

Interpretation of Discriminant Value

This discriminant value will tell us the position of the line relative to the circle. The concept is simple:

1. Positive discriminant ($\Delta > 0$) means the quadratic equation has two different real roots. In geometric context, this means the line intersects the circle at two points.

2. Zero discriminant ($\Delta = 0$) means the quadratic equation has one real root (repeated root). Geometrically, the line is tangent to the circle at one point.

3. Negative discriminant ($\Delta < 0$) means the quadratic equation has no real roots. Geometrically, the line does not intersect the circle.

Calculation Example

Let's look at a concrete example to make it clearer. Say we have line $y = 2x - 1$ and circle $x^2 + y^2 - 4x + 2y - 4 = 0$.

We substitute $y = 2x - 1$ into the circle equation:

From the quadratic equation $5x^2 - 4x - 5 = 0$, we identify its coefficients: $a = 5$, $b = -4$, and $c = -5$.

Calculate the discriminant:

Since $\Delta = 116 > 0$, the line $y = 2x - 1$ intersects the circle at two points.

This discriminant value not only tells us the position of the line, but also shows how many intersection points there are. The larger the positive discriminant value, the "farther" the line is from the tangent condition.

Standard Circle Case

For circles with center at the origin like $x^2 + y^2 = r^2$ and line $y = mx + c$, the process becomes more concise.

Substitute the line into the circle:

The discriminant for this case is:

After simplification:

The interpretation remains the same based on the sign of the discriminant.

Practice

1. Determine the position of line $y = x + 3$ relative to circle $x^2 + y^2 = 8$.

2. Investigate the position of line $y = -2x + 5$ relative to circle $x^2 + y^2 - 6x + 4y + 9 = 0$.

3. Line $2x + y - 4 = 0$ and circle $x^2 + y^2 = 5$. What is their position relationship?

4. Determine the value of $k$ so that line $y = x + k$ is tangent to circle $x^2 + y^2 = 18$.

Answer Key

1. Solution:

   Substitute $y = x + 3$ into $x^2 + y^2 = 8$:

   $$x^2 + (x + 3)^2 = 8$$

   $$x^2 + x^2 + 6x + 9 = 8$$

   $$2x^2 + 6x + 1 = 0$$

   Discriminant: $\Delta = 6^2 - 4(2)(1) = 36 - 8 = 28$

   Since $\Delta = 28 > 0$, the line intersects the circle at two points.

2. Solution:

   Substitute $y = -2x + 5$ into $x^2 + y^2 - 6x + 4y + 9 = 0$:

   $$x^2 + (-2x + 5)^2 - 6x + 4(-2x + 5) + 9 = 0$$

   $$x^2 + 4x^2 - 20x + 25 - 6x - 8x + 20 + 9 = 0$$

   $$5x^2 - 34x + 54 = 0$$

   Discriminant: $\Delta = (-34)^2 - 4(5)(54) = 1156 - 1080 = 76$

   Since $\Delta = 76 > 0$, the line intersects the circle at two points.

3. Solution:

   Change line $2x + y - 4 = 0$ to $y = -2x + 4$.

   Substitute into $x^2 + y^2 = 5$:

   $$x^2 + (-2x + 4)^2 = 5$$

   $$x^2 + 4x^2 - 16x + 16 = 5$$

   $$5x^2 - 16x + 11 = 0$$

   Discriminant: $\Delta = (-16)^2 - 4(5)(11) = 256 - 220 = 36$

   Since $\Delta = 36 > 0$, the line intersects the circle at two points.

4. Solution:

   For the line to be tangent to the circle, the discriminant must equal zero.

   Substitute $y = x + k$ into $x^2 + y^2 = 18$:

   $$x^2 + (x + k)^2 = 18$$

   $$x^2 + x^2 + 2kx + k^2 = 18$$

   $$2x^2 + 2kx + (k^2 - 18) = 0$$

   Discriminant: $\Delta = (2k)^2 - 4(2)(k^2 - 18) = 4k^2 - 8k^2 + 144 = -4k^2 + 144$

   For $\Delta = 0$:

   $$-4k^2 + 144 = 0$$

   $$4k^2 = 144$$

   $$k^2 = 36$$

   $$k = \pm 6$$

   Therefore the value is $k = 6$ or $k = -6$.