Pole and Polar of a Parabola
Pole and Polar of a Parabola
Understanding the Basics:
 Parabola Definition: A parabola is a type of conic section defined as the set of all points in a plane that are equidistant from a fixed line (the directrix) and a fixed point (the focus) not on the line.
Pole and Polar Definitions:

Pole: In the context of a parabola, the pole is a point from which perpendiculars to the tangent lines of the parabola are drawn. The pole is usually located outside the parabola.

Polar: The polar is the line that passes through the pole and is perpendicular to the tangent line of the parabola at a specific point. This line is a direct counterpart to the tangent line at that point.
Key Concepts:

PolePolar Relationship: For any point on a parabola, there's a unique polar line that passes through the pole and is perpendicular to the tangent at that point.

Properties of Pole and Polar:
 The polar of the focus of the parabola is the directrix.
 The polar of a point on the parabola is the tangent line at that point.
 The pole lies on the axis of the parabola and is equidistant from the focus and directrix.
 If a point lies on the polar of another point, the second point lies on the polar of the first point.
Equations:

Standard Equation of Parabola: ${y}^{2}=4ax$ (for a parabola with focus at $(a,0)$ and directrix $x=a$)

Pole and Polar Equation Relationship: For a point $(h,k)$ on the parabola, the equation of the tangent line is $y=mx+\frac{a}{m}$. Then, the equation of the polar line through $(h,k)$ (with respect to the pole $(p,q)$ is $yk=\frac{1}{m}(xh)$.
Steps to Find Pole and Polar:

Identify the Parabola's Equation: Ensure the equation of the parabola is in the standard form.

Determine the Point on the Parabola: Given a point on the parabola, $P(h,k)$, and a pole, $Q(p,q)$, determine the equation of the tangent at $P$.

Use the Perpendicularity Property: Use the fact that the polar is perpendicular to the tangent at $P$ to find the equation of the polar line.

Verify the Relationships: Check if the properties of pole and polar hold true for the given parabola and points.
Example:
Given: Consider the parabola with equation ${y}^{2}=4x$ and a point on the parabola, $P(4,4)$. Find the pole and polar of this parabola with respect to $P$.
Solution:
Step 1: Equation of the Parabola: ${y}^{2}=4x$
Step 2: Point on the Parabola: $P(4,4)$
Step 3: Finding the Equation of Tangent at $P(4,4)$:
Given the point $P(h,k)=(4,4)$ on the parabola, differentiate the equation ${y}^{2}=4x$ implicitly to get the derivative dy/dx:
$\frac{dy}{dx}=\frac{1}{2}\frac{1}{\sqrt{x}}$
At $x=4$, $\frac{dy}{dx}=\frac{1}{4}$
Thus, the slope of the tangent at $P$ is $m=\frac{1}{4}$.
The equation of the tangent at $P$ is: $y4=\frac{1}{4}(x4)$
Now, Finding the Pole and Polar:
Step 4: Finding the Pole (Q):
Given the point $P(4,4)$, the pole Q lies on the axis of the parabola and is equidistant from the focus and the directrix.
For the parabola ${y}^{2}=4x$, the focus is $(a,0)=(1,0)$ and the directrix is $x=a=1$.
The distance between the focus and the directrix is $2a=2\times 1=2$.
Thus, the pole $Q$ lies at a distance of $2$ units from the focus, i.e., $Q(3,0)$.
Step 5: Finding the Equation of the Polar Line:
The equation of the tangent at $P$ was found to be $y4=\frac{1}{4}(x4)$.
Now, the equation of the polar line passing through $P$ with respect to the pole $Q(3,0)$ will be perpendicular to this tangent.
The slope of the polar line is the negative reciprocal of the slope of the tangent, so the slope of the polar line is ${m}_{\text{polar}}=4$.
Using the pointslope form with the point $P(4,4)$, the equation of the polar line is: $y4=4(x4)$
Simplify the equation to get the final form of the polar line.
So, the polar line equation becomes: $y=4x+20$
Conclusion:
The pole of the parabola ${y}^{2}=4x$ with respect to the point $P(4,4)$ is $Q(3,0)$, and the equation of the polar line passing through $P$ with respect to $Q$ is $y=4x+20$.