Equation of tangent

Understanding the Equation of a Tangent to a Parabola

The equation of a tangent to a curve at a given point is a straight line that just touches the curve at that point. This line has the same slope as the curve at the point of tangency. When the curve is a parabola, the equation of the tangent can be derived using calculus or geometric properties.

Slope-Point Form of the Tangent Line

The slope-point form of the equation of a line is given by:

$$ y - y_1 = m(x - x_1) $$

where $(x_1, y_1)$ is a point on the line, and $m$ is the slope of the line. To find the equation of a tangent to a parabola, we need to know the slope of the parabola at the point of tangency.

Finding the Slope of the Parabola

For a parabola with the equation $y = ax^2 + bx + c$, the slope at any point is given by the derivative of $y$ with respect to $x$. The derivative is:

$$ \frac{dy}{dx} = 2ax + b $$

At the point of tangency $(x_1, y_1)$, the slope of the tangent line is therefore $m = 2ax_1 + b$.

Equation of the Tangent Line

Using the slope-point form and the slope found above, the equation of the tangent line to the parabola $y = ax^2 + bx + c$ at the point $(x_1, y_1)$ is:

$$ y - y_1 = (2ax_1 + b)(x - x_1) $$

Special Case: Parabola $y^2 = 4ax$

For the standard parabola $y^2 = 4ax$, the equation of the tangent at the point $(x_1, y_1)$ can be written in different forms:

  1. Slope form: If the slope of the tangent is $m$, the equation is $y = mx + a/m$.
  2. Point form: If the tangent touches the parabola at $(at^2, 2at)$, the equation is $y = tx - at^2$.
  3. Parametric form: Using the parametric coordinates $(at^2, 2at)$, the equation is $y = tx - at^3$.

Table of Differences and Important Points

Aspect General Parabola $y = ax^2 + bx + c$ Standard Parabola $y^2 = 4ax$
Slope at $(x_1, y_1)$ $m = 2ax_1 + b$ $m = \frac{4a}{y_1}$
Equation of Tangent $y - y_1 = (2ax_1 + b)(x - x_1)$ $y = mx + \frac{a}{m}$
Point of Tangency $(x_1, y_1)$ $(at^2, 2at)$
Parametric Form Not commonly used $y = tx - at^3$

Examples

Example 1: General Parabola

Find the equation of the tangent to the parabola $y = x^2 - 4x + 3$ at the point $(2, -1)$.

Solution:

  1. Calculate the slope of the parabola at $(2, -1)$: $$ m = 2ax + b = 2 \cdot 1 \cdot 2 - 4 = 0 $$

  2. Use the slope-point form: $$ y - (-1) = 0 \cdot (x - 2) $$

  3. Simplify to get the equation of the tangent: $$ y = -1 $$

Example 2: Standard Parabola

Find the equation of the tangent to the parabola $y^2 = 4x$ at the point $(1, 2)$.

Solution:

  1. Use the point form for the standard parabola: $$ y = tx - at^2 $$

  2. Since the point $(1, 2)$ lies on the parabola, $2^2 = 4 \cdot 1$, we have $a = 1$ and $t = 1$.

  3. Substitute $a$ and $t$ into the equation: $$ y = x - 1 $$

Understanding the equation of a tangent to a parabola is crucial for solving problems in calculus and analytic geometry. By knowing the form of the parabola and the point of tangency, one can derive the equation of the tangent line using the methods outlined above.