Finding the equation of a tangent to a circle is a fundamental concept in coordinate geometry. This guide provides a straightforward strategy, breaking down the process into manageable steps, ensuring you master this crucial topic. We'll focus on understanding the underlying principles and applying them effectively.
Understanding the Fundamentals: Circles and Tangents
Before diving into the equation, let's refresh our understanding of key concepts:
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Circle Equation: The standard equation of a circle with center (h, k) and radius r is:
(x - h)² + (y - k)² = r² -
Tangent: A tangent to a circle is a straight line that touches the circle at exactly one point, called the point of tangency. At the point of tangency, the tangent line is perpendicular to the radius of the circle. This perpendicularity is the key to finding the tangent equation.
Step-by-Step Strategy: Finding the Tangent Equation
Let's outline a clear, step-by-step approach to finding the equation of a tangent to a circle:
Step 1: Identify the Circle's Center and Radius
Begin by clearly identifying the center (h, k) and the radius (r) of the given circle equation. For example, if the equation is (x - 2)² + (y + 1)² = 9, then (h, k) = (2, -1) and r = 3.
Step 2: Determine the Point of Tangency
You'll typically be given the point of tangency (x₁, y₁). This point lies on both the circle and the tangent line. Ensure this point satisfies the circle's equation; this serves as a valuable check.
Step 3: Find the Slope of the Radius
The slope (m_radius) of the radius connecting the center (h, k) and the point of tangency (x₁, y₁) is calculated using the formula:
m_radius = (y₁ - k) / (x₁ - h)
Step 4: Determine the Slope of the Tangent
Since the tangent is perpendicular to the radius at the point of tangency, the product of their slopes equals -1. Therefore, the slope of the tangent (m_tangent) is:
m_tangent = -1 / m_radius
Important Note: If the radius is vertical (x₁ = h), the tangent will be horizontal, and its slope will be 0. If the radius is horizontal (y₁ = k), the tangent will be vertical, and its slope is undefined (represented by a vertical line x = x₁).
Step 5: Apply the Point-Slope Form
Now, use the point-slope form of a line equation to find the equation of the tangent:
y - y₁ = m_tangent * (x - x₁)
Substitute the values of y₁, x₁, and m_tangent you calculated in the previous steps.
Step 6: Simplify the Equation
Finally, simplify the equation to obtain the equation of the tangent line in the desired form (e.g., slope-intercept form: y = mx + c, or standard form: Ax + By + C = 0).
Example Problem: Putting it all Together
Let's find the equation of the tangent to the circle (x - 3)² + (y + 2)² = 16 at the point (5, 0).
- Center and Radius: (h, k) = (3, -2), r = 4
- Point of Tangency: (x₁, y₁) = (5, 0)
- Slope of Radius: m_radius = (0 - (-2)) / (5 - 3) = 1
- Slope of Tangent: m_tangent = -1 / 1 = -1
- Point-Slope Form: y - 0 = -1(x - 5)
- Simplified Equation: y = -x + 5
Therefore, the equation of the tangent line is y = -x + 5.
Mastering the Tangent Equation: Practice and Refinement
This straightforward approach will equip you with the skills to confidently tackle finding tangent equations. The more you practice with different examples, varying circle equations and points of tangency, the more proficient you'll become. Remember to meticulously follow each step and regularly check your work. Consistent practice is the key to mastering this important geometric concept.
