At what point in the first quadrant does the line with equation y= 0.5x + 3
 intersect the circle with radius 3 and center (0, 3)?


We are tasked with finding the point in the first quadrant where the line \( y = 0.5x + 3 \) intersects the circle with radius 3 and center at \( (0, 3) \). The equation of the circle is derived from the standard form of a circle, which is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

where \( (h, k) \) is the center, and \( r \) is the radius. For our circle, the center is \( (0, 3) \) and the radius is 3, so the equation of the circle becomes:

\[
x^2 + (y - 3)^2 = 9
\]

### Step 1: Substitute the equation of the line into the equation of the circle
We substitute \( y = 0.5x + 3 \) into the equation of the circle:

\[
x^2 + ((0.5x + 3) - 3)^2 = 9
\]

Simplifying the equation:

\[
x^2 + (0.5x)^2 = 9
\]

\[
x^2 + 0.25x^2 = 9
\]

\[
1.25x^2 = 9
\]

### Step 2: Solve for \( x \)
\[
x^2 = \frac{9}{1.25} = 7.2
\]

\[
x = \sqrt{7.2} \approx 2.683
\]

### Step 3: Find the corresponding \( y \)-value
Now, substitute \( x \approx 2.683 \) into the equation of the line \( y = 0.5x + 3 \):

\[
y = 0.5(2.683) + 3 \approx 1.341 + 3 = 4.341
\]

### Conclusion:
The point of intersection in the first quadrant is approximately \( (2.683, 4.341) \).

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. What is the general formula for the intersection of a line and a circle?
2. How can we determine if a line intersects a circle at more than one point?
3. What is the geometric significance of a circle's center lying on a line?
4. How would the solution change if the radius of the circle increased?
5. How would the intersection points change if the slope of the line were steeper?

**Tip**: When solving for intersections of curves, always verify if multiple solutions exist by considering the discriminant of the quadratic equation formed.

At what point in the first quadrant does the line with equation y = 0.5x + 3 intersect the circle with radius 3 and center (0, 3)?

Find the point of intersection between the line y = 0.5x + 3 and the circle with radius 3 centered at (0, 3). Learn step-by-step how to solve this coordinate geometry problem, involving substitution and solving quadratic equations. Suitable for students in Grades 10-12.

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