In the figure, the line 2t-by +9=0 cuts the circle at A and B on the .x-axis and y-axis respectively. Find the coordinates of A and B.

Given that the line \(2t - by + 9 = 0\) cuts the circle at points \(A\) and \(B\), where \(A\) lies on the \(x\)-axis and \(B\) lies on the \(y\)-axis, we need to find the coordinates of these points.

### Step 1: Understanding the Geometry

- Since \(A\) lies on the \(x\)-axis, its \(y\)-coordinate will be 0.
- Since \(B\) lies on the \(y\)-axis, its \(x\)-coordinate will be 0.

### Step 2: Equation of the Line

The equation of the line is given as:

\[
2t - by + 9 = 0
\]

We can rearrange this equation to find the points \(A\) and \(B\):

\[
by = 2t + 9
\]
\[
y = \frac{2t + 9}{b}
\]

### Step 3: Finding Point A (on the \(x\)-axis)

At point \(A\), \(y = 0\). Substituting \(y = 0\) in the line equation:

\[
2t - b(0) + 9 = 0
\]
\[
2t + 9 = 0
\]
\[
t = -\frac{9}{2}
\]

So, the coordinates of \(A\) are \(\left(-\frac{9}{2}, 0\right)\).

### Step 4: Finding Point B (on the \(y\)-axis)

At point \(B\), \(x = 0\). To find the \(y\)-coordinate, we use the equation of the line:

\[
2t + 9 = by
\]
\[
y = \frac{2t + 9}{b}
\]

Substituting \(x = 0\):

\[
B = \left(0, \frac{2t + 9}{b}\right)
\]

Now, substitute the value of \(t\) we found earlier:

\[
y = \frac{2\left(-\frac{9}{2}\right) + 9}{b} = \frac{-9 + 9}{b} = 0
\]

Thus, the coordinates of \(B\) are \(\left(0, 0\right)\).

However, since \(B\) lies on the \(y\)-axis and not at the origin, this implies a contradiction unless the constant \(t\) or \(b\) is adjusted appropriately based on the specific circle's equation, which wasn't provided. We need more details about the circle's equation or correct interpretations.

If you have more details or if there's a specific equation for the circle, please provide them, and I can refine the solution.

Would you like more details or have any other questions? Here are 5 related questions:

1. What is the general form of a circle's equation?
2. How do we find the intersection points of a line and a circle?
3. How can we verify if a point lies on a given line?
4. What role do the slope and intercept play in the equation of a line?
5. How can the equation of a circle help in finding points of intersection?

**Tip:** Always check the consistency of your points with the original geometric conditions or any provided constraints.

Learn how to find the coordinates of points A and B where the line 2t - by + 9 = 0 intersects a circle. This problem involves understanding the geometry of intersecting lines and circles, using equations of lines, and applying coordinate geometry concepts.

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