To find the points $A$ and $B$ where the line $3y = 4x - 15$ intersects the curve $8x^2 - 27y^2 = 45$, we need to solve the system of equations. Let's go step by step:

Step 1: Express $y$ from the line equation

The equation of the line is:

$3y = 4x - 15$

Solve for $y$:

$y = \frac{4x - 15}{3}$

Step 2: Substitute $y$ in the curve equation

Now substitute $y = \frac{4x - 15}{3}$ into the curve equation:

$8x^2 - 27y^2 = 45$

Substituting for $y$:

$8x^2 - 27\left(\frac{4x - 15}{3}\right)^2 = 45$

Step 3: Simplify the equation

First, square the expression for $y$:

$\left(\frac{4x - 15}{3}\right)^2 = \frac{(4x - 15)^2}{9}$

So the equation becomes:

$8x^2 - 27 \cdot \frac{(4x - 15)^2}{9} = 45$

Simplify the fraction:

$8x^2 - 3 \cdot (4x - 15)^2 = 45$

Step 4: Expand $(4x - 15)^2$

Expand $(4x - 15)^2$:

$(4x - 15)^2 = 16x^2 - 120x + 225$

Substitute this into the equation:

$8x^2 - 3(16x^2 - 120x + 225) = 45$

Step 5: Simplify the equation

Distribute the 3:

$8x^2 - (48x^2 - 360x + 675) = 45$

Simplify further:

$8x^2 - 48x^2 + 360x - 675 = 45$

Combine like terms:

$-40x^2 + 360x - 675 = 45$

Subtract 45 from both sides:

$-40x^2 + 360x - 720 = 0$

Step 6: Solve the quadratic equation

Divide the equation by -40 to simplify:

$x^2 - 9x + 18 = 0$

Now solve this quadratic equation using the quadratic formula:

$x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(18)}}{2(1)}$

$x = \frac{9 \pm \sqrt{81 - 72}}{2}$

$x = \frac{9 \pm \sqrt{9}}{2}$

$x = \frac{9 \pm 3}{2}$

So the two solutions for $x$ are:

$x = \frac{9 + 3}{2} = 6 \quad \text{or} \quad x = \frac{9 - 3}{2} = 3$

Step 7: Find the corresponding $y$ values

Now substitute these values of $x$ into the equation for $y = \frac{4x - 15}{3}$:

• For $x = 6$:

$y = \frac{4(6) - 15}{3} = \frac{24 - 15}{3} = \frac{9}{3} = 3$

• For $x = 3$:

$y = \frac{4(3) - 15}{3} = \frac{12 - 15}{3} = \frac{-3}{3} = -1$

Step 8: Conclusion

The points of intersection are:

$A(6, 3) \quad \text{and} \quad B(3, -1)$

Would you like any further explanation or more details?

Here are 5 related questions for further practice:

1. How do you solve systems of equations with a line and a circle?
2. Can you find the distance between points $A(6, 3)$ and $B(3, -1)$?
3. What happens if the line does not intersect the curve?
4. How do you solve a system with a line and a parabola?
5. How does the discriminant help determine the nature of the roots in quadratic equations?

Tip: For quadratic equations, always check the discriminant ($b^2 - 4ac$) to determine whether the equation has real, imaginary, or repeated roots.