Step 1: Understanding the Concept:
This problem explores the geometric properties of a cubic curve, specifically the interaction between its tangents and intersection points. When a tangent is drawn to a cubic curve at one point, it generally intersects the curve at exactly one other point. This sequence of points generates a pattern in their x-coordinates (abscissae).
Step 2: Key Formula or Approach:
1. Equation of tangent at \( P_1(x_1, y_1) \): \( y - y_1 = m(x - x_1) \), where \( m = \frac{dy}{dx} \).
2. Solve the cubic equation \( x^3 - y_{tangent} = 0 \) to find the other intersection point \( x_2 \).
Step 3: Detailed Explanation:
Let the point on the curve \( y = x^3 \) be \( P_1(x_1, x_1^3) \).
The derivative of the curve is \( \frac{dy}{dx} = 3x^2 \).
The slope of the tangent at \( P_1 \) is \( m_1 = 3x_1^2 \).
The equation of the tangent at \( P_1 \) is:
\[ y - x_1^3 = 3x_1^2(x - x_1) \implies y = 3x_1^2 x - 2x_1^3 \]
To find where this tangent meets the curve \( y = x^3 \) again, we equate the two:
\[ x^3 = 3x_1^2 x - 2x_1^3 \]
\[ x^3 - 3x_1^2 x + 2x_1^3 = 0 \]
We know that \( x = x_1 \) is a root of this equation with multiplicity 2 (since the line is tangent at \( x_1 \)). Thus, \( (x - x_1)^2 \) must be a factor.
Factoring the cubic:
\[ (x - x_1)^2 (x + 2x_1) = 0 \]
The roots are \( x = x_1 \) (tangency point) and \( x = -2x_1 \) (intersection point).
Therefore, the abscissa of \( P_2 \) is \( x_2 = -2x_1 \).
By repeating this process for \( P_2 \), the tangent at \( P_2 \) will intersect the curve at \( P_3 \) with abscissa:
\[ x_3 = -2x_2 = -2(-2x_1) = 4x_1 = (-2)^2 x_1 \]
In general, for any point \( P_n \), the abscissa is \( x_n = x_1 (-2)^{n-1} \).
This is a Geometric Progression (G.P.) where the first term is \( x_1 \) and the common ratio is \( -2 \).
Step 4: Final Answer:
The abscissae form a G.P. with a common ratio of -2.