https://file.aiquickdraw.com/mathbot/file/1725127599310knl1q7rr.jpg

(a) In the graph above, the curve (*y*+1)^2−*x^*2=8 meets the straight line 3*x*+2*y*=−5 at point *P* and *Q*. Find the coordinates of point *P* and *Q*.
Coordinates of point *P* = (?,?)
Coordinates of point *Q* = (?,?)
(b) The straight line x3*x*+2*y*=−5 meets another line x*y*−2*x*=1 at point *R*. Find the coordinates of point *R*. Coordinates of point *R* = (?,?)

### (a) Finding the Coordinates of Points P and Q

The problem involves finding the points of intersection between the curve \((y + 1)^2 - x^2 = 8\) and the straight line \(3x + 2y = -5\).

#### Step 1: Express \(y\) from the line equation
We start by solving the line equation for \(y\):

\[
3x + 2y = -5 \implies 2y = -3x - 5 \implies y = -\frac{3}{2}x - \frac{5}{2}
\]

#### Step 2: Substitute into the curve equation
Next, substitute this expression for \(y\) into the curve equation \((y + 1)^2 - x^2 = 8\):

\[
\left(-\frac{3}{2}x - \frac{5}{2} + 1\right)^2 - x^2 = 8
\]

Simplifying the expression inside the square:

\[
y + 1 = -\frac{3}{2}x - \frac{3}{2} \implies \left(-\frac{3}{2}(x + 1)\right)^2 - x^2 = 8
\]

Simplifying the square:

\[
\left(\frac{9}{4}(x + 1)^2\right) - x^2 = 8
\]

Expanding and simplifying:

\[
\frac{9}{4}(x^2 + 2x + 1) - x^2 = 8 \implies \frac{9}{4}x^2 + \frac{9}{2}x + \frac{9}{4} - x^2 = 8
\]

Multiply by 4 to eliminate the fraction:

\[
9x^2 + 18x + 9 - 4x^2 = 32
\]

\[
5x^2 + 18x - 23 = 0
\]

#### Step 3: Solve the quadratic equation
We solve the quadratic equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 5\), \(b = 18\), and \(c = -23\):

\[
x = \frac{-18 \pm \sqrt{18^2 - 4(5)(-23)}}{2(5)} = \frac{-18 \pm \sqrt{324 + 460}}{10} = \frac{-18 \pm \sqrt{784}}{10} = \frac{-18 \pm 28}{10}
\]

This gives us two possible values for \(x\):

\[
x_1 = 1 \quad \text{and} \quad x_2 = -4.6
\]

#### Step 4: Find corresponding \(y\) values
Substitute these \(x\) values back into \(y = -\frac{3}{2}x - \frac{5}{2}\):

For \(x_1 = 1\):

\[
y_1 = -\frac{3}{2}(1) - \frac{5}{2} = -\frac{3}{2} - \frac{5}{2} = -\frac{8}{2} = -4
\]

For \(x_2 = -4.6\):

\[
y_2 = -\frac{3}{2}(-4.6) - \frac{5}{2} = \frac{13.8}{2} - \frac{5}{2} = \frac{8.8}{2} = 4.4
\]

So, the coordinates of \(P\) and \(Q\) are:

\[
\text{Coordinates of } P = (1, -4) \quad \text{and} \quad Q = (-4.6, 4.4)
\]

### (b) Finding the Coordinates of Point R

Now, we find the intersection of the lines \(3x + 2y = -5\) and \(y - 2x = 1\).

#### Step 1: Express \(y\) from the second line equation

\[
y = 2x + 1
\]

#### Step 2: Substitute into the first line equation

Substitute \(y = 2x + 1\) into \(3x + 2y = -5\):

\[
3x + 2(2x + 1) = -5 \implies 3x + 4x + 2 = -5 \implies 7x = -7 \implies x = -1
\]

#### Step 3: Find the corresponding \(y\) value

Substitute \(x = -1\) into \(y = 2x + 1\):

\[
y = 2(-1) + 1 = -2 + 1 = -1
\]

Thus, the coordinates of \(R\) are:

\[
\text{Coordinates of } R = (-1, -1)
\]

---

**Summary of Results:**

- Coordinates of point **P** = \((1, -4)\)
- Coordinates of point **Q** = \((-4.6, 4.4)\)
- Coordinates of point **R** = \((-1, -1)\)

Would you like to see any specific details or have any questions on this solution?

### Related Questions:
1. How do you determine the number of intersection points between a line and a curve?
2. Can a quadratic equation have complex solutions, and if so, what does that mean geometrically?
3. How does the discriminant of a quadratic equation affect the nature of its roots?
4. What are some methods to solve a system of linear equations geometrically?
5. How can you interpret the intersection of two conic sections in a plane?

### Tip:
When solving systems of equations involving curves, substitution is a powerful method, but always simplify expressions carefully to avoid errors in the subsequent algebraic steps.

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