https://file.aiquickdraw.com/mathbot/file/1725993476385t1suh3ho.jpg

Let's break down the problem from the image:

### Problem:
- The x-axis is a tangent to a circle with center \((-2, 4)\).
- There are two sub-parts:
  1. What is the length of the radius of the circle?
  2. Write down the equation of the circle.

---

### Solution:

#### (i) Length of the Radius:

The distance from the center of the circle \((-2, 4)\) to the x-axis is the length of the radius because the x-axis is tangent to the circle.

- The distance from a point \((x_1, y_1)\) to the x-axis is given by the absolute value of the y-coordinate, i.e., \(|y_1|\).
- Here, the center is \((-2, 4)\), so the radius is the absolute value of the y-coordinate:
  \[
  r = |4| = 4
  \]
  
Thus, the radius of the circle is **4 units**.

#### (ii) Equation of the Circle:

The general 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.

- The center of the circle is \((-2, 4)\), so \(h = -2\) and \(k = 4\).
- From part (i), we know that the radius \(r = 4\).

Therefore, the equation of the circle is:
\[
(x + 2)^2 + (y - 4)^2 = 4^2
\]
Simplifying:
\[
(x + 2)^2 + (y - 4)^2 = 16
\]

---

### Final Answers:
1. The length of the radius of the circle is **4 units**.
2. The equation of the circle is \((x + 2)^2 + (y - 4)^2 = 16\).

---

Do you want any further details or have questions? 

Here are some related questions you might find useful:
1. How do you find the equation of a circle when given two points on its circumference?
2. What is the equation of a tangent to a circle at a given point?
3. How would the equation change if the circle were tangent to the y-axis?
4. How can we calculate the area of the circle from its equation?
5. What is the distance between two circles if their centers are known?

**Tip:** Always remember that the distance from a point to the x-axis is simply the absolute value of the y-coordinate of that point!

The x-axis is a tangent to the circle with centre (-2, 4). (i) What is the length of the radius of the circle? (ii) Write down the equation of the circle.

Solve a geometry problem involving a circle with center (-2, 4) tangent to the x-axis. Learn how to calculate the radius and find the equation of the circle using key geometric concepts and formulas. Suitable for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation