the line 3x = 4x-15 intersects the curve 8x^2 - 27y^2 = 45 at point a and b find a and b

We are given two equations:

1. A line equation:  
   \[
   3x = 4x - 15
   \]

2. A hyperbola equation:  
   \[
   8x^2 - 27y^2 = 45
   \]

### Step 1: Simplify the line equation.
Rearrange the line equation \( 3x = 4x - 15 \):

\[
3x - 4x = -15
\]

\[
-x = -15
\]

\[
x = 15
\]

### Step 2: Substitute \( x = 15 \) into the hyperbola equation.
Now that we know \( x = 15 \), substitute it into the hyperbola equation \( 8x^2 - 27y^2 = 45 \):

\[
8(15^2) - 27y^2 = 45
\]

Calculate \( 15^2 = 225 \):

\[
8(225) - 27y^2 = 45
\]

\[
1800 - 27y^2 = 45
\]

### Step 3: Solve for \( y^2 \).
Rearrange the equation to isolate \( y^2 \):

\[
1800 - 45 = 27y^2
\]

\[
1755 = 27y^2
\]

\[
y^2 = \frac{1755}{27} = 65
\]

Thus,

\[
y = \pm \sqrt{65}
\]

### Step 4: Find points A and B.
Since we have \( x = 15 \) and \( y = \pm \sqrt{65} \), the points of intersection \( A \) and \( B \) are:

\[
A(15, \sqrt{65}) \quad \text{and} \quad B(15, -\sqrt{65})
\]

### Final Answer:
The points of intersection are \( A(15, \sqrt{65}) \) and \( B(15, -\sqrt{65}) \).

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Would you like more details or have any questions?

Here are 5 related questions to extend your learning:

1. How would the solution change if the hyperbola equation was different?
2. What if the line equation had a different slope?
3. Can you find the distance between points A and B?
4. How would you graph the intersection of a line and a hyperbola?
5. What are the general methods to find intersections between conic sections and lines?

**Tip**: When solving systems of equations involving a line and a curve, always try to substitute one variable into the other equation after simplifying the line equation.

The line 3x = 4x - 15 intersects the curve 8x^2 - 27y^2 = 45 at points A and B. Find A and B.

Find the points of intersection between the line 3x = 4x - 15 and the hyperbola 8x^2 - 27y^2 = 45 with this step-by-step solution. This problem demonstrates key algebraic concepts such as solving systems of equations and the properties of hyperbolas. Suitable for students in Grades 10-12.

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