How to Find the Condition for a Line to be Tangent to a Circle

Today, we are going to look at an essential lesson for any SAT student: understanding the condition for a line to be tangent to a circle.

It sounds technical, but it’s actually a beautiful geometric moment where a line and a circle meet at exactly one point. In the SAT world, “tangent” is just code for “perfection.” Here is a breakdown of how to find that condition so you can snag those extra points on test day.

What Does “Tangent” Actually Mean?

When we say a line is tangent to a curve, we mean they touch at exactly one point. Think of it like a high-five between a straight line and a curve; they meet, they touch, but they don’t cross over each other or 

• A tangent to a circle is the line that touches the circle at only one point. There can be only one tangent at a point to a circle. The point of tangency is the point at which the tangent meets the circle.
• A circle can have many tangents but at a particular point on the circumference of the circle, only one tangent passes through that point on the circle. The tangent to a circle is always perpendicular to the radius of the circle.

Condition for Tangency

1. Geometric Approach:

The most intuitive way to think about this is by looking at the radius.

• The Rule: A line is tangent to a circle if the perpendicular distance from the center of the circle to the line is exactly equal to the radius.
• Why it works: If the distance to the line is shorter than the radius, the line is inside the circle. If it’s longer, the line is floating away in space. When they are equal, you’ve found your tangent.

2. Distance Approach:

Calculate the perpendicular distance d from the circle center( \(\displaystyle (x_{0}, y_{0})\) ) to the line Ax + By + C = 0 using \(\displaystyle d = \frac{|A x_{0} + B y_{0} + C|}{\sqrt{A^{2} + B^{2}}}\)
The line is tangent if d = r.

3. The Algebraic Way (The “Discriminant” Method)

• The Step: Substitute the equation of the line (e.g., y = mx + c) into the equation of the circle.
• The Result: You will end up with a quadratic equation.
• The Condition: For the line to be tangent, there must be exactly one solution. In algebra, this means the discriminant (b² – 4ac = 0) must equal zero.

Substitute the line equation (y = mx + c) into the circle equation (x² + y² = a²). Rearrange into a quadratic equation: Ax² + Bx + C = 0.

The line is tangent if the discriminant D = B² – 4AC = 0.

Specific Formulae:

1. For x² + y² = a² and y = mx + c: The condition is c² = a²(1 + m²)
2. The equation of the tangent line is \(\displaystyle y = mx \pm a\sqrt{1 + m^{2}}\)
3. for general Circle (x – h)² + (y-k)² = r² and Ax + By + C = 0. the condition is \(\displaystyle \frac{|Ah + Bk + C|}{\sqrt{A^{2} + B^{2}}} = r\)
   
   

For a more detailed walkthrough, you can watch this video: