Tangent Line Equation: Circle At (2,4) Explained

Hey guys! Let's dive into a cool math problem: figuring out the equation of a tangent line to a circle. Specifically, we'll tackle finding the tangent line at the point (2, 4) on the circle defined by the equation $(x+4)^2 + (y-5)^2 = 37$. This might sound a bit intimidating at first, but trust me, it's totally manageable. We'll break it down into easy-to-follow steps, so you can ace this. Ready? Let's get started!

Understanding the Basics: Circles and Tangent Lines

Alright, before we jump into the calculations, let's make sure we're all on the same page. A circle is a set of all points equidistant from a central point. That distance is called the radius. The standard form equation of a circle is $(x-h)^2 + (y-k)^2 = r^2$, where (h, k) is the center of the circle, and r is the radius. Now, what's a tangent line? It's a line that touches the circle at exactly one point, and it's perpendicular to the radius at that point of contact. Knowing this relationship between the radius and the tangent line is key to solving our problem. Think of it like this: the radius goes straight out from the center of the circle to the point where the tangent line just kisses the circle's edge. They form a perfect right angle.

In our case, the circle's equation is $(x+4)^2 + (y-5)^2 = 37$. Comparing this to the standard form, we can identify the center of the circle as (-4, 5) and the radius squared is 37 (so the radius is the square root of 37, but we won't need that directly). The point (2, 4) is the point on the circle where we want to find the tangent line. Our goal is to find the equation of the line that touches the circle at (2, 4) at a right angle to the radius drawn from the center (-4, 5) to (2, 4).

To make the process even smoother, it's super helpful to sketch a quick diagram. This visual aid can help you see the relationships between the center, the point on the circle, the radius, and the tangent line. It doesn't have to be perfect; just a rough sketch can be a game-changer in understanding the geometry involved. Remember, understanding the geometry helps a lot in determining the equation of the tangent line for the circle. So, take a moment to draw a simple circle, mark the center, plot the point (2, 4), and roughly sketch what the tangent line would look like. That visual representation is super important!

Step-by-Step: Calculating the Tangent Line Equation

Now, let's get down to the actual calculations. Here's the step-by-step process to find the equation of the tangent line:

1. Find the slope of the radius: We need to find the slope of the radius connecting the center of the circle (-4, 5) to the point (2, 4). The slope formula is $(y_2 - y_1) / (x_2 - x_1)$. Plugging in our points, we get $(4 - 5) / (2 - (-4)) = -1 / 6$. So, the slope of the radius is -1/6.

2. Find the slope of the tangent line: Since the tangent line is perpendicular to the radius, their slopes are negative reciprocals of each other. If the slope of the radius is -1/6, then the slope of the tangent line is 6 (because -1 / (-1/6) = 6). Remember that the product of slopes of perpendicular lines is always -1. This understanding is very important for the calculation.

3. Use the point-slope form: Now that we have the slope of the tangent line (6) and a point on the line (2, 4), we can use the point-slope form of a linear equation: $y - y_1 = m(x - x_1)$. Plugging in our values, we get $y - 4 = 6(x - 2)$. This form allows us to go straight to the line equation. The beauty of point-slope form is its directness: all you need is a point and a slope to find the equation of a line.

4. Convert to slope-intercept form (optional): To make the equation look more familiar, let's convert it to slope-intercept form (y = mx + b). Expand the equation we found in step 3: $y - 4 = 6x - 12$. Add 4 to both sides: $y = 6x - 8$. Voila! The equation of the tangent line is $y = 6x - 8$.

So, by carefully applying the geometric relationships and basic algebra, we've successfully determined the equation of the tangent line. This equation represents a line that touches the circle at the point (2, 4) at a right angle to the radius. Knowing the standard forms of both the circle and line is very important.

Verification and Interpretation

To make sure we've got it right, let's do a quick check. Does the point (2, 4) actually lie on the line $y = 6x - 8$? Plugging in x = 2, we get $y = 6(2) - 8 = 12 - 8 = 4$. Yep, it checks out! This confirms that our tangent line does indeed pass through the point (2, 4). Verification is a critical part of math. Always double-check your work to make sure your answer makes sense.

What does this equation, $y = 6x - 8$, really mean? It tells us everything about the tangent line. The slope of 6 indicates how steeply the line rises as you move to the right. The y-intercept of -8 tells us where the line crosses the y-axis. The equation is a complete description of the tangent line, which is perpendicular to the circle's radius at the point (2, 4).

By going through these steps, you can see that finding the equation of a tangent line isn't as complex as it might initially appear. It's all about understanding the basic properties of circles and tangent lines, using the slope formula, and applying the point-slope form. It's like a puzzle, and each step brings you closer to the complete picture.

Practice Makes Perfect

Want to become a tangent line pro? Here's a tip: practice, practice, practice! Try solving similar problems with different circles and points. Change the equation or vary the point on the circle. The more you practice, the more comfortable you'll become with the process. You can also try to find the equation of the normal line to the circle at the point. Always try to think about the geometry of the problem – visualize the circle, the center, the point, and the tangent line.

Remember, math is like any skill: the more you work at it, the better you get. Don't be discouraged if you don't get it right away. Keep practicing, and you'll be solving these problems with confidence in no time. Also, don't hesitate to check your answers with online calculators or textbooks, especially when you're starting out. Seeing how others solve the problem can be a great learning tool. Always remember that the most important part is understanding the concepts and enjoying the process!