Circle Tangent To A Line Find The Radius

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Hey guys! Today, we're diving into a classic math problem involving circles and lines. Specifically, we're going to figure out how to find the radius of a circle that's tangent to a given line. This type of problem pops up quite often in analytical geometry, so understanding the underlying concepts is super important. Let's break it down step-by-step!

Understanding the Problem

Our problem states that we have a circle defined by the equation (x - 1)² + (y - 3)² = r². This tells us a couple of key things right off the bat. First, the center of the circle is at the point (1, 3). Remember, the general equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Second, the problem also gives us a line: 5x + 12y = 66. The crucial piece of information is that the circle is tangent to this line. What does that mean, exactly? Tangency means the line touches the circle at only one point. This single point of contact gives us a vital geometric relationship that we can use to solve for the radius.

Now, the main question: what is the value of r? We are given five options: (a) √10, (b) 25/13, (c) 13/12, (d) 60/13, and (e) 19/13. Our mission is to use the information about the circle's equation and the line's equation, along with the tangency condition, to determine which of these options is the correct radius. To nail this, we'll use a formula that relates the distance from a point to a line, a key concept in analytical geometry. Let's dive into the steps to solve this!

The Key Concept: Distance from a Point to a Line

The distance from a point to a line is the shortest distance between the point and any point on the line. This shortest distance is always along the perpendicular line segment connecting the point to the line. In our case, the point is the center of the circle (1, 3), and the line is 5x + 12y = 66. Here's the kicker: because the circle is tangent to the line, the distance from the center of the circle to the line is exactly equal to the radius r of the circle. This is the linchpin of our solution!

There's a handy formula to calculate the distance (d) from a point (x₀, y₀) to a line given by the equation Ax + By + C = 0:

d = |Ax₀ + By₀ + C| / √(A² + B²)

This formula might look a bit intimidating at first, but it's actually pretty straightforward to use. Let's break down each part:

  • |…|: The vertical bars indicate the absolute value. This means we only care about the magnitude of the result, not the sign. Distances are always positive.
  • Ax₀ + By₀ + C: This is the equation of the line with the coordinates of the point plugged in. We'll calculate this value first.
  • √(A² + B²): This part involves the coefficients A and B from the line's equation. We square each coefficient, add them together, and then take the square root. This gives us a normalizing factor related to the line's slope.

Now, let's apply this formula to our problem. First, we need to rewrite the equation of the line 5x + 12y = 66 in the form Ax + By + C = 0. Subtracting 66 from both sides, we get:

5x + 12y - 66 = 0

So, we have A = 5, B = 12, and C = -66. The point (x₀, y₀) is the center of our circle, which is (1, 3). Now we have all the pieces we need to plug into the formula.

Applying the Formula and Solving for r

Let's plug the values into the distance formula:

d = |(5)(1) + (12)(3) - 66| / √(5² + 12²)

Now, let's simplify:

d = |5 + 36 - 66| / √(25 + 144) d = |-25| / √169 d = 25 / 13

Remember, the distance d we just calculated is equal to the radius r of the circle because the circle is tangent to the line. So, we've found that r = 25/13.

Looking back at our answer choices, we see that option (b) is 25/13. Therefore, the correct answer is (b).

Wrapping Up and Key Takeaways

Guys, we nailed it! We successfully found the radius of the circle by using the distance from a point to a line formula and the crucial understanding that the distance from the center of a circle to a tangent line is equal to the circle's radius. This is a common type of problem in analytical geometry, and mastering it will definitely help you out. Remember: The key is to connect the geometric interpretation of tangency with the algebraic tools we have at our disposal, like the distance formula. Always visualize the problem if you can – it can make understanding the relationships much easier. The distance from a point to a line formula is a powerful tool that can be applied in many different situations. So, keep practicing, and you'll become a pro at these problems in no time!

Practice Problems

To solidify your understanding, try these similar problems:

  1. Find the radius of the circle (x + 2)² + (y - 1)² = r² that is tangent to the line 3x - 4y + 7 = 0.
  2. A circle has its center at (4, -2) and is tangent to the line x + y = 5. Determine the radius of the circle.
  3. What is the radius of the circle x² + y² = r² that is tangent to the line y = x + 2?

Work through these, and you'll be a master of circle-tangent problems in no time! Good luck, and keep learning!