How to Find the Pedal Equation of a Polar Curve

TL;DR

To find the pedal equation of the polar curve defined by r = 2(1 + cos θ), start by differentiating the logarithm of the equation with respect to θ. This leads to the expression cot φ = -tan(θ/2). Using the formula for the perpendicular distance from the tangent to the pole, you can derive the pedal equation p³ = 4p².

Transcript

hello everyone in this session we'll discuss one problem on pedal equation of a polar curve so the equation is r equal to 2 times of 1 plus cos theta for which we have to find pedal equation now starting with the given equation that is r equal to 2 times of 1 plus cos theta let us take log on both sides so we'll have log of r equal to log 2 plus lo... Read More

Key Insights

• 🇨🇷 The equation r = 2(1 + cos θ) can be transformed into the pedal equation p^3 = 4p^2.
• 💈 The pedal equation helps find the perpendicular distance of the tangent to the curve from the pole.
• 🖐️ The angle φ plays a significant role in the derivation of the pedal equation.
• 😑 The tangent of π/2 minus θ/2 can be expressed as cot φ.

Questions & Answers

Q: What is the equation for a polar curve in the video?

The equation for the polar curve is r = 2(1 + cos θ).

Q: How is the pedal equation derived?

The pedal equation is derived by taking the logarithm of the polar curve equation and differentiating with respect to θ.

Q: What is the significance of φ in the pedal equation?

φ represents the angle between the tangent line and the x-axis and is crucial in determining the perpendicular distance of the tangent to the curve from the pole.

Q: How can the equation p^3 = 4p^2 be understood?

The equation p^3 = 4p^2 is the simplified version of the pedal equation, representing the relationship between the perpendicular distance p and the radius r.

Summary & Key Takeaways

• The video explores the equation r = 2(1 + cos θ) and how to find its pedal equation.

• By taking the logarithm of both sides and differentiating with respect to θ, the equation can be simplified to cot φ = -tan(θ/2).

• Using the perpendicular distance formula (p = r*sin φ), the equation is further derived to p^3 = 4p^2.