3y=4x-12 intersects the curve 8x^2 - 27y^2= 45 find the points of intersection a and b

To find the points of intersection between the line \( 3y = 4x - 12 \) and the curve \( 8x^2 - 27y^2 = 45 \), we need to solve these equations simultaneously.

### Step 1: Simplify the equation of the line
The line equation is given by:
\[
3y = 4x - 12
\]
Divide by 3:
\[
y = \frac{4}{3}x - 4
\]

### Step 2: Substitute \( y \) in the curve equation
Substitute \( y = \frac{4}{3}x - 4 \) into the curve equation:
\[
8x^2 - 27y^2 = 45
\]

Replacing \( y \):
\[
8x^2 - 27\left(\frac{4}{3}x - 4\right)^2 = 45
\]

### Step 3: Expand and simplify
Let's simplify the equation by expanding the squared term:
\[
\left(\frac{4}{3}x - 4\right)^2 = \left(\frac{4}{3}x\right)^2 - 2 \cdot \frac{4}{3}x \cdot 4 + 4^2
\]
\[
= \frac{16}{9}x^2 - \frac{32}{3}x + 16
\]

Substituting into the curve equation:
\[
8x^2 - 27\left(\frac{16}{9}x^2 - \frac{32}{3}x + 16\right) = 45
\]

Simplify further:
\[
8x^2 - \left( \frac{432}{9}x^2 - \frac{864}{3}x + 432 \right) = 45
\]
\[
8x^2 - 48x^2 + 288x - 432 = 45
\]
\[
-40x^2 + 288x - 432 = 45
\]
\[
-40x^2 + 288x - 477 = 0
\]

Now solve this quadratic equation for \( x \). Let's solve for \( x \) using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where:
\[
a = -40, \quad b = 288, \quad c = -477
\]

### Step 4: Find the values of \( x \)

Let's calculate the discriminant \( b^2 - 4ac \) and the values of \( x \).The solutions for \( x \) are:

\[
x_1 \approx 2.58 \quad \text{and} \quad x_2 \approx 4.62
\]

### Step 5: Find corresponding \( y \) values
Now, substitute these \( x \) values back into the equation of the line \( y = \frac{4}{3}x - 4 \) to find the corresponding \( y \) values.

Let's calculate the corresponding \( y \) values for \( x_1 \) and \( x_2 \).The points of intersection between the line \( 3y = 4x - 12 \) and the curve \( 8x^2 - 27y^2 = 45 \) are:

- Point A: \( (x_1, y_1) \approx (2.58, -0.56) \)
- Point B: \( (x_2, y_2) \approx (4.62, 2.16) \)

Would you like more details or have any questions about these calculations?

Here are some related questions to consider:

1. How do you find the points of intersection between two curves generally?
2. What is the significance of solving quadratic equations in finding intersections?
3. How can you verify the points of intersection graphically?
4. What methods can be used to solve quadratic equations if the discriminant is negative?
5. How do changes in the coefficients of the line affect the points of intersection?

**Tip:** When solving simultaneous equations involving curves and lines, substituting one equation into the other can help reduce the problem to a simpler equation to solve.

3y=4x-12 intersects the curve 8x^2 - 27y^2= 45. Find the points of intersection a and b.

Learn how to solve a system of equations involving the line 3y=4x-12 and the curve 8x^2 - 27y^2=45. This problem includes key algebraic concepts like solving quadratic equations and using the quadratic formula to find points of intersection. Suitable for students in Grades 10-12.

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