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: y2=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+am. Then, the equation of the polar line through (h,k) (with respect to the pole (p,q) is yk=1m(xh).

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(h,k), and a pole, Q(p,q), 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 y2=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: y2=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 y2=4x implicitly to get the derivative dy/dx:

dydx=121x

At x=4, dydx=14

Thus, the slope of the tangent at P is m=14.

The equation of the tangent at P is: y4=14(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 y2=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=2×1=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=14(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 mpolar=4.

Using the point-slope 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 y2=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.