Tangent Lines To Circles: Find The Equations Easily

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Hey guys! Today, we're diving into the exciting world of circles and tangents. We're going to solve a couple of problems that involve finding the equations of tangent lines to circles. Don't worry, we'll break it down step by step so it's super easy to follow. Let's get started!

Exercise 3: Finding the Tangent Line to (x + 1)² + y² = 20 at (-3, 4)

In this first exercise, our mission is to find the equation of the tangent line to the circle defined by the equation (x + 1)² + y² = 20. We know that the point where this tangent line touches the circle is (-3, 4). So, how do we tackle this? Here's the breakdown:

  1. Understanding the Circle's Equation:

    • The general form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. In our case, the equation is (x + 1)² + y² = 20. This means our circle has a center at (-1, 0) and a radius of √20 (which simplifies to 2√5).

  2. Finding the Slope of the Radius:

    • The tangent line is perpendicular to the radius at the point of tangency. So, first, we need to find the slope of the radius connecting the center of the circle (-1, 0) and the point (-3, 4). The slope (m) is calculated as (y₂ - y₁) / (x₂ - x₁). Plugging in the values, we get m = (4 - 0) / (-3 - (-1)) = 4 / -2 = -2. So, the slope of the radius is -2. This is a crucial step, as it sets the stage for finding the tangent line's slope.

  3. Finding the Slope of the Tangent Line:

    • Since the tangent line is perpendicular to the radius, its slope is the negative reciprocal of the radius's slope. If the slope of the radius is -2, the slope of the tangent line (mₜ) is -1 / -2 = 1/2. Boom! We've got the slope of our tangent line. This perpendicular relationship is key in solving these types of problems.

  4. Using the Point-Slope Form:

    • Now that we have the slope of the tangent line (1/2) and a point it passes through (-3, 4), we can use the point-slope form of a linear equation: y - y₁ = m(x - x₁). Plugging in the values, we get y - 4 = (1/2)(x - (-3)), which simplifies to y - 4 = (1/2)(x + 3).

  5. Simplifying to Slope-Intercept Form (Optional):

    • If we want to express the equation in slope-intercept form (y = mx + b), we can further simplify: y - 4 = (1/2)x + 3/2. Adding 4 to both sides gives us y = (1/2)x + 3/2 + 4, which simplifies to y = (1/2)x + 11/2. So, our tangent line equation is y = (1/2)x + 11/2. It is important to know how to manipulate the equation into different forms.

Therefore, the equation of the tangent line to the circle (x + 1)² + y² = 20 at the point (-3, 4) is y = (1/2)x + 11/2.

Exercise 4: Determining the Tangent Line to x² + y² - 8x - 4y = 0 at (8, 4)

Alright, let's move on to the second exercise. This time, we need to determine the equation of the tangent line to the circle defined by x² + y² - 8x - 4y = 0 at the point (8, 4). This looks a bit different, right? The equation isn't in the standard form we saw earlier. No sweat, we'll handle it!

  1. Completing the Square to Find the Circle's Center and Radius:

    • The given equation is x² + y² - 8x - 4y = 0. To find the center and radius, we need to complete the square for both x and y terms. Rearrange the equation: (x² - 8x) + (y² - 4y) = 0.
    • To complete the square for x² - 8x, we take half of the coefficient of x (-8), square it ((-4)² = 16), and add it to both sides. Similarly, for y² - 4y, we take half of the coefficient of y (-4), square it ((-2)² = 4), and add it to both sides. This gives us: (x² - 8x + 16) + (y² - 4y + 4) = 0 + 16 + 4.
    • Now we can rewrite the equation as (x - 4)² + (y - 2)² = 20. Aha! Now it's in the standard form. We can see that the center of the circle is (4, 2) and the radius is √20 (or 2√5). Completing the square is essential for extracting key information from the circle's equation.

  2. Finding the Slope of the Radius:

    • We need to find the slope of the radius connecting the center (4, 2) and the point (8, 4). Using the slope formula, m = (y₂ - y₁) / (x₂ - x₁), we get m = (4 - 2) / (8 - 4) = 2 / 4 = 1/2. So, the slope of the radius is 1/2. Remember, the radius is the line segment connecting the center of the circle to the point of tangency.

  3. Finding the Slope of the Tangent Line:

    • The tangent line is perpendicular to the radius, so its slope is the negative reciprocal of the radius's slope. If the slope of the radius is 1/2, the slope of the tangent line (mₜ) is -1 / (1/2) = -2. We've found the slope of the tangent line: -2. The negative reciprocal is the key to finding the perpendicular slope.

  4. Using the Point-Slope Form:

    • Now that we have the slope of the tangent line (-2) and a point it passes through (8, 4), we use the point-slope form: y - y₁ = m(x - x₁). Plugging in the values, we get y - 4 = -2(x - 8).

  5. Simplifying to Slope-Intercept Form (Optional):

    • To express the equation in slope-intercept form (y = mx + b), we simplify: y - 4 = -2x + 16. Adding 4 to both sides gives us y = -2x + 20. Therefore, the equation of our tangent line is y = -2x + 20. Knowing different forms of the linear equation can be helpful.

Therefore, the equation of the tangent line to the circle x² + y² - 8x - 4y = 0 at the point (8, 4) is y = -2x + 20.

Key Takeaways

  • Understanding the Circle's Equation: Knowing the standard form of a circle's equation, (x - h)² + (y - k)² = r², helps you quickly identify the center (h, k) and radius (r).
  • Completing the Square: If the circle's equation isn't in standard form, complete the square to rewrite it.
  • Perpendicular Slopes: The tangent line is always perpendicular to the radius at the point of tangency. This means their slopes are negative reciprocals of each other.
  • Point-Slope Form: Use the point-slope form of a linear equation (y - y₁ = m(x - x₁)) to find the equation of the tangent line, given its slope and a point it passes through.

By following these steps, you can confidently find the equations of tangent lines to circles! Keep practicing, and you'll become a pro in no time. Happy solving!