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We are given a circle centered at the origin with a radius of \( \sqrt{145} \), and we need to find the coordinates of the two points on the circle where \( x = 9 \).

### Step 1: Equation of the Circle

The equation of a circle centered at the origin \((0, 0)\) with radius \( r \) is:

\[
x^2 + y^2 = r^2
\]

Here, the radius is \( \sqrt{145} \), so the equation becomes:

\[
x^2 + y^2 = 145
\]

### Step 2: Substitute \( x = 9 \)

We are asked to find the points where \( x = 9 \), so we substitute \( x = 9 \) into the equation of the circle:

\[
9^2 + y^2 = 145
\]

\[
81 + y^2 = 145
\]

### Step 3: Solve for \( y^2 \)

\[
y^2 = 145 - 81
\]

\[
y^2 = 64
\]

### Step 4: Solve for \( y \)

Taking the square root of both sides:

\[
y = \pm 8
\]

### Step 5: Coordinates of the Points

The coordinates of the two points where \( x = 9 \) are:

\[
(9, 8) \quad \text{and} \quad (9, -8)
\]

Thus, the two points on the circle where \( x = 9 \) are \( (9, 8) \) and \( (9, -8) \).

Would you like further details or explanations on any steps? Here are some related questions to explore further:

1. How do we derive the general equation of a circle?
2. Why does solving for \( y \) give two possible values?
3. What is the geometrical significance of the radius \( \sqrt{145} \)?
4. How can you find the coordinates where \( y = k \) instead of \( x = 9 \)?
5. How would the problem change if the circle were centered at a different point?

**Tip:** When solving for \( y^2 \) or any squared variable, remember to consider both the positive and negative square roots.

A circle is centered at the origin and has a radius of √145. Work out the coordinates of the two points on the circle where x = 9.

Learn how to calculate the coordinates of two points on a circle with a radius of √145, centered at the origin, when x = 9. This problem explores circle equations and algebraic techniques. Suitable for students in Grades 9-11.

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