Family of Parabolas Tangent to a Line
The family of parabolas tangent to the line \( r \), with equation \( y = mx + q \), at the point \( T(x_T,y_T) \) is described by the equation
$$ y = mx + q + k(x - x_T)^2 $$
In this expression, \( k \) is a parameter. By changing its value, we obtain different parabolas, all of which touch the same line at the same point.
An Example
To see how this works, consider the line
$$ y = -x + 2 $$
and the point \( T(1,1) \), which lies on the line.
The corresponding family of tangent parabolas is
$$ y = -x + 2 + k(x - 1)^2 $$
Every choice of \( k \) produces a different parabola. Despite their different shapes, all of them remain tangent to the line at the point \( T \).
First parabola: \( k = 1 \)
Substituting \( k = 1 \) into the equation gives
$$ y = -x + 2 + (x - 1)^2 $$
Expanding and simplifying, we obtain
$$ y = -x + 2 + x^2 - 2x + 1 $$
$$ y = x^2 - 3x + 3 $$
The graph below shows this parabola. It touches the line exactly at the point \( T \).
Second parabola: \( k = -1 \)
Now let \( k = -1 \).
$$ y = -x + 2 - (x - 1)^2 $$
$$ y = -x + 2 - x^2 + 2x - 1 $$
$$ y = -x^2 + x + 1 $$
This parabola opens downward, yet it is still tangent to the same line at the same point.
How the Parameter \( k \) Affects the Parabola
The parameter \( k \) controls the shape and orientation of the parabola:
- If \( k > 0 \), the parabola opens upward.
- If \( k < 0 \), the parabola opens downward.
- Larger values of \( |k| \) produce narrower parabolas.
- Smaller values of \( |k| \) produce wider parabolas.
For example:
- \( k = 1 \) gives \( y = x^2 - 3x + 3 \).
- \( k = -1 \) gives \( y = -x^2 + x + 1 \).
- \( k = 2 \) gives \( y = 2x^2 - 5x + 4 \).
Although these parabolas have different equations and different shapes, they all belong to the same family because they share the same tangency condition.
In this example, every parabola is tangent to the line \( y = -x + 2 \) at the point \( T(1,1) \).
Why the Formula Works
We now show why the equation
$$ y = mx + q + k(x - x_T)^2 $$
always generates parabolas tangent to the line \( r \) at the point \( T(x_T,y_T) \).
Start with the line
$$ r:\; y = mx + q $$
and let \( T(x_T,y_T) \) be a point on the line. Since \( T \) belongs to \( r \), its coordinates satisfy
$$ y_T = mx_T + q $$
Now consider the family of curves
$$ y = mx + q + k(x - x_T)^2 $$
To prove that each curve is tangent to the line at \( T \), we need to verify two facts:
- The point \( T \) lies on the parabola.
- The parabola and the line have the same slope at \( T \).
Checking the Slope
Differentiate the equation with respect to \( x \):
$$ \frac{dy}{dx} = m + 2k(x - x_T) $$
Evaluating the derivative at \( x = x_T \) gives
$$ \left.\frac{dy}{dx}\right|_{x=x_T} = m + 2k(x_T - x_T) = m $$
This is exactly the slope of the line \( r \). Therefore, the parabola and the line share the same tangent direction at the point \( T \).
Checking the Point of Tangency
Substituting \( x_T \) into the equation of the parabola gives
$$ y = mx_T + q + k(x_T - x_T)^2 $$
$$ y = mx_T + q $$
Since \( y_T = mx_T + q \), the point \( T \) lies on the parabola.
Conclusion
The equation
$$ y = mx + q + k(x - x_T)^2 $$
generates an infinite family of parabolas that all pass through the point \( T(x_T,y_T) \) and are tangent to the line \( r: y = mx + q \) at that point.
Changing the parameter \( k \) changes the shape of the parabola, but not its point of tangency or its tangent line. This is precisely what defines the family of parabolas tangent to a given line at a fixed point.
