# 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:

1. Pole-Polar 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.

2. 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 $\left(a,0\right)$ and directrix $x=-a$)

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

#### Steps to Find Pole and Polar:

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

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

3. 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.

4. 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, $P\left(4,4\right)$. 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: $P\left(4,4\right)$

Step 3: Finding the Equation of Tangent at $P\left(4,4\right)$:

Given the point $P\left(h,k\right)=\left(4,4\right)$ 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: $y-4=\frac{1}{4}\left(x-4\right)$

#### Now, Finding the Pole and Polar:

Step 4: Finding the Pole (Q):

Given the point $P\left(4,4\right)$, 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 $\left(a,0\right)=\left(1,0\right)$ and the directrix is $x=-a=-1$.

The distance between the focus and the directrix is $2a=2×1=2$.

Thus, the pole $Q$ lies at a distance of $2$ units from the focus, i.e., $Q\left(3,0\right)$.

Step 5: Finding the Equation of the Polar Line:

The equation of the tangent at $P$ was found to be $y-4=\frac{1}{4}\left(x-4\right)$.

Now, the equation of the polar line passing through $P$ with respect to the pole $Q\left(3,0\right)$ 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 point-slope form with the point $P\left(4,4\right)$, the equation of the polar line is: $y-4=-4\left(x-4\right)$

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 $P\left(4,4\right)$ is $Q\left(3,0\right)$, and the equation of the polar line passing through $P$ with respect to $Q$ is $y=-4x+20$.