Tangent Line Equation: Circle (1,-4), Radius 10, Point (-5,4)

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Let's dive into finding the equation of a tangent line to a circle! This is a classic geometry problem that combines circle properties and linear equations. We'll break it down step-by-step to make it super clear. So, stick with me, guys, and we'll conquer this together!

Understanding the Problem

First, let's recap what we're dealing with. We have a circle. Think of it like a perfectly round pizza! This circle has a center, which is the point right in the middle, kind of like where you'd place that first perfect slice. In our case, the center is at the coordinates (1, -4). The circle also has a radius, which is the distance from the center to any point on the edge of the circle – like the crust of our pizza! Here, the radius is 10 units.

Now, imagine a line that just barely touches the circle at one point. This line is called a tangent line. It's like a special guest that only makes a brief appearance. We know this tangent line touches the circle at the point (-5, 4). Our mission is to find the equation of this tangent line. Think of it like finding the secret code to unlock this line's identity.

To find the equation, we will need to use a couple of key geometric principles. Firstly, we'll use the fact that the tangent line to a circle is always perpendicular to the radius at the point of tangency. This means the line connecting the center of the circle to the point where the tangent touches forms a right angle (90 degrees) with the tangent line. Secondly, we'll use the point-slope form of a linear equation to construct the equation of the tangent line. So, let's gear up and get started!

Steps to Find the Tangent Line Equation

1. Find the Slope of the Radius

Okay, first things first, we need to figure out the slope of the line segment that connects the center of the circle (1, -4) to the point of tangency (-5, 4). This line segment is essentially the radius of the circle that extends to the point where the tangent line kisses the circle.

The slope is a measure of how steep a line is. Think of it like climbing a hill – a steeper hill has a higher slope. Mathematically, we calculate the slope (often denoted as m) using the formula:

m = (y₂ - y₁) / (x₂ - x₁)

Where (x₁, y₁) and (x₂, y₂) are the coordinates of two points on the line. In our case:

(x₁, y₁) = (1, -4) (center of the circle) (x₂, y₂) = (-5, 4) (point of tangency)

Plugging these values into our slope formula, we get:

m = (4 - (-4)) / (-5 - 1) = 8 / -6 = -4/3

So, the slope of the radius is -4/3. Keep this number in your back pocket, we'll need it for the next step!

2. Find the Slope of the Tangent Line

Now, here's a crucial piece of information: the tangent line is perpendicular to the radius at the point of tangency. Perpendicular lines have slopes that are negative reciprocals of each other. What does that mean in plain English? It means we flip the fraction and change the sign.

If the slope of the radius is -4/3, then the slope of the tangent line (let's call it m_tangent) is:

m_tangent = -1 / (-4/3) = 3/4

Awesome! We've found the slope of our tangent line. We're halfway there!

3. Use the Point-Slope Form of a Line

Alright, we've got the slope of the tangent line (3/4), and we know a point that lies on the tangent line: (-5, 4). Now we can use the point-slope form of a linear equation to write the equation of the tangent line. The point-slope form is:

y - y₁ = m(x - x₁)

Where:

  • m is the slope of the line
  • (x₁, y₁) is a point on the line

Plugging in our values:

  • m = 3/4
  • (x₁, y₁) = (-5, 4)

We get:

y - 4 = (3/4)(x - (-5))

y - 4 = (3/4)(x + 5)

4. Convert to Standard Form (Ax + By + C = 0)

Okay, we've got the equation in point-slope form, which is great! But to match the answer choices, we need to convert it to standard form, which looks like this:

Ax + By + C = 0

Where A, B, and C are constants.

Let's do some algebraic maneuvering. First, get rid of the fraction by multiplying both sides of the equation by 4:

4(y - 4) = 4 * (3/4)(x + 5)

4y - 16 = 3(x + 5)

Now, distribute the 3 on the right side:

4y - 16 = 3x + 15

To get it into standard form, we want all the terms on one side, so let's subtract 3x and 15 from both sides:

4y - 16 - 3x - 15 = 0

Rearrange the terms to match the standard form (Ax + By + C = 0):

-3x + 4y - 31 = 0

To make the coefficient of x positive, we can multiply the entire equation by -1:

3x - 4y + 31 = 0

5. Matching the options

After reviewing the original answer choices: 6x - 8y - 62 = 0, 8y - 6x - 62 = 0, 6x - 8y + 62 = 0, 8y - 6x + 62 = 0. If we multiply our derived equation 3x - 4y + 31 = 0 by 2, we get 6x - 8y + 62 = 0, which exactly matches one of the options.

The Answer

So, the equation of the tangent line g is 6x - 8y + 62 = 0.

Key Takeaways

Let's recap the main ideas we've used in this problem:

  • Tangent to a circle: A tangent line touches the circle at only one point.
  • Perpendicularity: The radius drawn to the point of tangency is perpendicular to the tangent line.
  • Slope Formula: m = (y₂ - y₁) / (x₂ - x₁)
  • Perpendicular Slopes: Perpendicular lines have slopes that are negative reciprocals of each other.
  • Point-Slope Form: y - y₁ = m(x - x₁)
  • Standard Form: Ax + By + C = 0

By understanding these concepts and applying them systematically, you can tackle a wide range of circle and tangent line problems. Keep practicing, and you'll become a geometry whiz in no time!

So there you have it, guys! We successfully found the equation of the tangent line. Remember to break down the problem into smaller, manageable steps, and don't be afraid to use the formulas and concepts you've learned. You got this!