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.