Understanding the Tangent Lines of a Parabola

The concept of a tangent line to a parabola is a foundational idea in both geometry and calculus. In this article, we delve into the method of finding the equation of tangents to a parabola given specific points through which the lines must pass. We will specifically tackle the parabola given by the equation y - 22 -12x 1 and the point (-1, -1).

Equation of the Parabola

Starting with the given parabola equation:

y - 22 -12x 1

This equation represents a standard form of a parabola. To simplify the process of finding the tangent line, we will shift the origin to the point (2, -1). The new equation of the parabola in the shifted coordinate system will be:

y2 -12(x - 4)

Note that this implies: a -3, x' x - 1, and y' y - 2.

Finding the Tangent Line Equation

The equation of the tangent line in slope form y mx c can be derived as:

c u03B1/m where u03B1 a and thus c -3/m.

By substituting the given point (-1, -1) into the equation, we can solve for the slope m and subsequently, the equation of the tangent line. Let's walk through the steps:

Step 1: Determine the Slope

The coordinates of the point are (0, -3) when we revert to the original coordinate system, which gives us the equation:

-3 m0 - 3/m.

Solving for m yields:

m 1.

Thus, the equation of the tangent line in the original coordinate system is:

y - 2 1(x - 1) which simplifies to:

y x - 1 2 or y x 1.

Step 2: Verify the Point of Contact

Now, we need to find the point of contact of the tangent line with the parabola. Substituting y x 1 into the parabola equation (y - 2)2 -12x 1 gives us:

((x 1) - 2)2 -12x 1

Simplifying:

(x - 1)2 -12x 1.

Expanding and rearranging terms:

x2 - 2x 1 -12x 1

x2 1 0

x(x 10) 0

This yields two solutions: x 0 and x -10. Substituting these back into the line equation:

For x 0, y 1 -> Point: (0, 1)

For x -10, y -9 -> Point: (-10, -9)

Graphical Interpretation

However, we need to consider only the point that satisfies the requirement for tangency. From the given problem, we know that one of the tangents is at the vertex, which is (-1, 2). The other tangent, corresponding to the point (-4, -4), can be derived through the slope-intercept form. Both tangents are confirmed through the graph:

Image: Two tangent lines are shown, one at the vertex (-1, 2) and the other passing through (-1, -1) and (-4, -4).

Conclusion

The key points in this solution are understanding the relationship between the parabola equation and its tangent lines, and using algebraic methods to find the required tangents. This is a fundamental skill in calculus and analytic geometry, with applications in physics and engineering.