8 The line $$ 4 y-3 x=7 $$ meets the curve $$ 3 x^{2}-8 y=-5 $$ at points $$ P $$ and $$ Q $$. Find the gradient of the curve at $$ P $$ and at $$ Q $$

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To find the gradients of the curve $$ 3x^2 - 8y = -5 $$ at the points $$ P $$ and $$ Q $$ where it intersects the line $$ 4y - 3x = 7 $$, follow these steps:

### 1. Find the Points of Intersection $$ P $$ and $$ Q $$

**Given:**
- Line: $$ 4y - 3x = 7 $$  
  Solve for $$ y $$:
  $$
  y = \frac{3x + 7}{4}
  $$

- Curve: $$ 3x^2 - 8y = -5 $$

**Substitute $$ y $$ from the line into the curve:**
$$
3x^2 - 8\left(\frac{3x + 7}{4}\right) = -5
$$
$$
3x^2 - 2(3x + 7) = -5
$$
$$
3x^2 - 6x - 14 = -5
$$
$$
3x^2 - 6x - 9 = 0
$$
$$
x^2 - 2x - 3 = 0 \quad \text{(Divide by 3)}
$$

**Solve the quadratic equation:**
$$
x = \frac{2 \pm \sqrt{4 + 12}}{2} = \frac{2 \pm \sqrt{16}}{2} = \frac{2 \pm 4}{2}
$$
$$
x = 3 \quad \text{or} \quad x = -1
$$

**Find corresponding $$ y $$ values:**
- For $$ x = 3 $$:
  $$
  y = \frac{3(3) + 7}{4} = \frac{16}{4} = 4 \quad \Rightarrow \quad P = (3, 4)
  $$
  
- For $$ x = -1 $$:
  $$
  y = \frac{3(-1) + 7}{4} = \frac{4}{4} = 1 \quad \Rightarrow \quad Q = (-1, 1)
  $$

### 2. Find the Gradient $$ \frac{dy}{dx} $$ of the Curve

**Differentiate the curve implicitly:**
$$
3x^2 - 8y = -5
$$
$$
\frac{d}{dx}(3x^2) - \frac{d}{dx}(8y) = \frac{d}{dx}(-5)
$$
$$
6x - 8\frac{dy}{dx} = 0
$$
$$
\frac{dy}{dx} = \frac{6x}{8} = \frac{3x}{4}
$$

**Calculate the gradients at $$ P $$ and $$ Q $$:**
- At $$ P = (3, 4) $$:
  $$
  \frac{dy}{dx} = \frac{3(3)}{4} = \frac{9}{4}
  $$
  
- At $$ Q = (-1, 1) $$:
  $$
  \frac{dy}{dx} = \frac{3(-1)}{4} = -\frac{3}{4}
  $$

### **Final Answer:**
- The gradient at point $$ P $$ is $$ \frac{9}{4} $$.
- The gradient at point $$ Q $$ is $$ -\frac{3}{4} $$.

**Answer:**
After simplifying, the curve has slope 9⁄4 at P and –3⁄4 at Q. Thus,

• dy/dx at P is 9/4

• dy/dx at Q is –3/4
At point P, the gradient is 9⁄4, and at point Q, the gradient is –3⁄4.

8 The line meets the curve
at points and . Find
the gradient of the curve at and at

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Solución

1. Find the Points of Intersection and

2. Find the Gradient of the Curve

Final Answer:

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