Tangent Line Equation: Circle At Point (4,5)

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Alright guys, let's dive into the fascinating world of circles and tangent lines! Today, we're going to explore how to find the equation of a tangent line to a circle, specifically when that line passes through a given point. It might sound a bit intimidating at first, but trust me, we'll break it down step by step so you can conquer these types of problems with ease.

Understanding the Basics of Tangent Lines

So, what exactly is a tangent line? In simple terms, a tangent line is a line that touches a circle at only one point. Imagine a straight line just barely grazing the edge of a circular shape – that’s your tangent line. This point of contact is super important, and we call it the point of tangency. The crucial property of a tangent line is that it's always perpendicular to the radius of the circle drawn to the point of tangency. This perpendicularity is the key that unlocks many problems involving tangent lines.

When we talk about the equation of a tangent line, we're essentially trying to describe this line mathematically. Just like any other line, we can define a tangent line using various forms of equations, such as the slope-intercept form (y = mx + c) or the point-slope form (y - y1 = m(x - x1)). Our mission is to find the specific equation that represents the tangent line in our problem.

Key Concepts to Remember

Before we jump into solving the problem, let's quickly recap some essential concepts that will be our building blocks:

  • Circle Equation: The standard equation of a circle with center (h, k) and radius r is (x - h)² + (y - k)² = r².
  • Slope of a Line: The slope (m) of a line passing through points (x1, y1) and (x2, y2) is given by m = (y2 - y1) / (x2 - x1).
  • Perpendicular Lines: If two lines are perpendicular, the product of their slopes is -1. In other words, if one line has a slope of m1, the slope of a line perpendicular to it (m2) is given by m2 = -1 / m1.

Keep these concepts in your mental toolkit, and you'll be well-equipped to tackle tangent line problems.

Problem Setup: The Given Information

Okay, let's get down to the specific problem we're tackling today. We're given a scenario where we need to find the equation of a tangent line to a circle. We know that this tangent line passes through the point (4, 5). Additionally, we're told that the tangent line has the equation x + 3y - 45 = 0.

Now, this might seem like we already have the answer, but here’s the catch: The equation x + 3y - 45 = 0 is a tangent line, but it's not necessarily the only tangent line, and we need to make sure it's the one that fits our specific scenario. We need to verify if this line is indeed tangent to the circle we're working with and if the point (4, 5) lies on this line.

Visualizing the Problem

It often helps to visualize these problems. Imagine a circle, and then picture a line touching the circle at one point. This is our tangent line. Now, we have a specific point (4, 5) in mind. Our goal is to confirm if the line x + 3y - 45 = 0 is the tangent line that passes through this point. If it is, then we've got our answer! If not, we'll need to explore other possibilities or correct the given information.

Initial Checks and Verifications

Before we proceed with more complex calculations, let's do a quick check to see if the given information makes sense. First, we'll check if the point (4, 5) actually lies on the line x + 3y - 45 = 0. To do this, we simply substitute x = 4 and y = 5 into the equation:

4 + 3(5) - 45 = 4 + 15 - 45 = -26

Since the result is -26, which is not equal to 0, the point (4, 5) does not lie on the line x + 3y - 45 = 0. This tells us there might be an issue with the problem statement or the given equation of the tangent line. We'll need to address this discrepancy before we can find the correct equation.

Resolving the Discrepancy and Finding the Circle's Information

Okay, guys, we've hit a snag! The point (4, 5) doesn't lie on the given line x + 3y - 45 = 0. This means we need to re-evaluate our approach. There's likely some missing information or a slight error in the problem statement. To solve this, we need to figure out the circle's equation or at least its center and radius. Without knowing the circle, we can't definitively find the tangent line.

Let's assume, for the sake of moving forward, that the intended problem involved finding the tangent line to a circle with a specific center and radius, and that the line x + 3y - 45 = 0 was perhaps a distraction or a piece of information related to a different part of the problem. We'll need to make some assumptions and work through a more general method for finding tangent lines.

A General Approach: Finding the Tangent Line

Since we're missing crucial information about the circle, let's outline a general method for finding the tangent line to a circle when given a point outside the circle. This approach typically involves the following steps:

  1. Assume a general equation for the tangent line: We can start with the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the given point (4, 5) in our case. So, we have y - 5 = m(x - 4).
  2. Use the condition of tangency: A line is tangent to a circle if the perpendicular distance from the center of the circle to the line is equal to the radius of the circle. This is a critical geometric property.
  3. Find the perpendicular distance: We can use the formula for the perpendicular distance (d) from a point (h, k) to a line Ax + By + C = 0, which is given by d = |Ah + Bk + C| / √(A² + B²).
  4. Set up an equation: Equate the perpendicular distance to the radius (r) of the circle. This will give us an equation involving the slope (m) of the tangent line and the circle's parameters (center and radius).
  5. Solve for the slope: Solve the equation obtained in step 4 for the possible values of m. There will generally be two solutions, corresponding to the two tangent lines that can be drawn from a point outside the circle.
  6. Write the equations of the tangent lines: Substitute the values of m obtained in step 5 back into the point-slope form of the line to get the equations of the tangent lines.

Let's Make an Assumption About the Circle

To illustrate this method, let's assume the circle has a center at (0, 0) and a radius of 5. This is a common and simple circle equation (x² + y² = 25), which will make our calculations easier. Now, let's apply the steps outlined above.

Applying the General Method: A Worked Example

With our assumed circle equation x² + y² = 25 and the point (4, 5), let's walk through the steps to find the tangent line(s).

  1. Assume a general equation for the tangent line: Using the point-slope form with the point (4, 5), we have y - 5 = m(x - 4). Let's rewrite this in the general form Ax + By + C = 0: mx - y - 4m + 5 = 0.
  2. Use the condition of tangency: The perpendicular distance from the center (0, 0) to the tangent line must equal the radius, which is 5.
  3. Find the perpendicular distance: Using the formula for perpendicular distance, we have: d = |A(0) + B(0) + C| / √(A² + B²) = |m(0) - 1(0) - 4m + 5| / √(m² + (-1)²) d = |-4m + 5| / √(m² + 1)
  4. Set up an equation: Equate the perpendicular distance to the radius (5): |-4m + 5| / √(m² + 1) = 5
  5. Solve for the slope: Square both sides to get rid of the square root and absolute value: (-4m + 5)² / (m² + 1) = 25 (16m² - 40m + 25) = 25(m² + 1) 16m² - 40m + 25 = 25m² + 25 0 = 9m² + 40m 0 = m(9m + 40) This gives us two possible slopes: m = 0 and m = -40/9.
  6. Write the equations of the tangent lines:
    • For m = 0: y - 5 = 0(x - 4) => y = 5
    • For m = -40/9: y - 5 = (-40/9)(x - 4) => 9y - 45 = -40x + 160 => 40x + 9y - 205 = 0

So, for the assumed circle x² + y² = 25, the tangent lines passing through (4, 5) are y = 5 and 40x + 9y - 205 = 0.

Conclusion: The Importance of Clear Problem Statements

Guys, this exercise highlights the importance of having a clear and complete problem statement. When we started, the information seemed contradictory, and we had to make assumptions to move forward. In real-world scenarios, it's crucial to ensure that you have all the necessary information before attempting to solve a problem.

We explored the concept of tangent lines, the condition of tangency, and a general method for finding the equation of a tangent line to a circle from a point outside the circle. We also saw how assumptions and educated guesses can help when faced with incomplete information.

Remember, the key to mastering these concepts is practice! Work through various problems, and you'll become more comfortable with the techniques and the underlying geometry. Keep up the great work, and you'll be a tangent line pro in no time!