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### Problem Overview:
- We are given a point \( P \) on the curve \( y = x^3 \).
- The tangent line at point \( P \) intersects the curve again at another point \( Q \).
- We're tasked with finding certain information about this intersection.

### Part (a): Equation of the Tangent Line at \( P \)

Given that \( P \) has an \( x \)-coordinate of \( a \), we want to find the equation of the tangent line at \( P \).

- The function is \( y = x^3 \).
- The slope of the tangent line at any point is given by the derivative of \( y \) with respect to \( x \), so:
  \[
  \frac{dy}{dx} = 3x^2
  \]
  At \( P \), where \( x = a \), the slope \( m \) is:
  \[
  m = 3a^2
  \]
  The equation of a tangent line is generally given by:
  \[
  y - y_1 = m(x - x_1)
  \]
  At point \( P = (a, a^3) \), we plug in the values to get:
  \[
  y - a^3 = 3a^2(x - a)
  \]
  Simplifying:
  \[
  y = 3a^2x - 2a^3
  \]
  So, the equation of the tangent line is \( y = 3a^2x - 2a^3 \).

### Part (b): Intersection of the Tangent Line with the Curve

We are told that the tangent line intersects the curve \( y = x^3 \) at two points: one is \( P \), and we need to find the coordinates of the other point \( Q \).

- Set the equation of the tangent line equal to the curve's equation:
  \[
  3a^2x - 2a^3 = x^3
  \]
  Rearrange this into a cubic equation:
  \[
  x^3 - 3a^2x + 2a^3 = 0
  \]
  We know \( x = a \) is a solution (since \( P \) is on both the tangent line and the curve). So, we can factor \( (x - a) \) out of the cubic equation:
  \[
  (x - a)(x^2 + ax - 2a^2) = 0
  \]
  Now solve for the remaining factor \( x^2 + ax - 2a^2 = 0 \). Using the quadratic formula:
  \[
  x = \frac{-a \pm \sqrt{a^2 + 8a^2}}{2}
  \]
  Simplifying:
  \[
  x = \frac{-a \pm 3a}{2}
  \]
  So, \( x = a \) (which we already know) or \( x = -2a \).

Thus, the coordinates of the second intersection point \( Q \) are \( (-2a, -8a^3) \).

### Part (c): Ratio of the Slopes at \( P \) and \( Q \)

We are asked to find the ratio of the slope of the curve at \( Q \) to the slope at \( P \).

- The slope at any point on the curve is given by \( \frac{dy}{dx} = 3x^2 \).
  - At \( P \) (where \( x = a \)), the slope is \( 3a^2 \).
  - At \( Q \) (where \( x = -2a \)), the slope is:
    \[
    3(-2a)^2 = 3(4a^2) = 12a^2
    \]
  - The ratio of the slope at \( Q \) to the slope at \( P \) is:
    \[
    \frac{12a^2}{3a^2} = 4
    \]

Thus, the ratio is \( 4 \).

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Let me know if you would like any further clarifications! 

### 5 Related Questions:
1. How would the result change if the curve were \( y = x^n \) instead of \( y = x^3 \)?
2. What happens to the tangent line equation if \( P \) is at \( x = 0 \)?
3. How can you solve higher-degree polynomial intersections like this problem more efficiently?
4. What is the significance of the ratio of the slopes at different points of a curve geometrically?
5. How would this problem differ if we worked with an implicit curve equation like \( y^2 = x^3 \)?

### Tip:
Always check if the equation you're dealing with is factorable by known solutions to simplify your work. In this case, knowing \( x = a \) as a root helped us quickly factor the cubic equation.

Let P be a point on the graph of y = x^3. The tangent line at P will intersect the graph at the point Q. (a) If P has x-coordinate a, then what is the equation of the tangent line at P? (b) The line in (a) will intersect the curve y = x^3 in two places; one of these is P and the other point is Q. (c) What is the ratio of the slope of the curve at Q to the slope at P?

Explore how to determine the equation of the tangent line at a point P on the curve y = x^3 and find the intersection at Q. Includes step-by-step solutions for solving cubic equations, calculating slopes, and applying the quadratic formula. Suitable for high school calculus students.

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