Chapter 10: Q. 11 (page 700)

Identify the conic represented by the polar equation $r = \frac{3}{1 - 2 \cos θ}$ . Find the rectangular equation

Short Answer

Expert verified

The given equation is of Hyperbola and the required equation in rectangular form is $\frac{y^{2}}{3} - \frac{{(x + 2)}^{2}}{1} = 1$

Step by step solution

Step 1. Given Information

The given equation is $r = \frac{3}{1 - 2 \cos θ}$

Step 2. Calculation

As we know that the cartesian coordinate of a point is related with its polar coordinate as follows,

$x = r \cos θ, y = r \sin θ \to r = \sqrt{x^{2} + y^{2}}$

Substitute the values in the given equation as follows,

$r = \frac{3}{1 - 2 \cos θ} r - 2 r \cos θ = 3 \sqrt{x^{2} + y^{2}} - 2 x = 3 {(\sqrt{x^{2} + y^{2}})}^{2} = {(3 + 2 x)}^{2} x^{2} + y^{2} = 9 + 4 x^{2} + 12 x 3 x^{2} - y^{2} + 12 x + 9 = 0$

From the cartesian form of the given polar equation, we can say that the given equation is of Hyperbola.

Step 3. Calculation

Convert the cartesian form of hyperbolic equation in the general form,

$\frac{3 x^{2} - y^{2} + 12 x + 9}{3} = 0 x^{2} + 4 x - \frac{y^{2}}{3} = - 3 (x^{2} + 4 x + 2^{2}) - \frac{y^{2}}{3} = - 3 + 2^{2} {(x + 2)}^{2} - y$ ²3=-1y23-(x+2)21=1

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Identify the conic represented by the polar equation $r = \frac{3}{1 - 2 \cos θ}$ . Find the rectangular equation

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

Step 3. Calculation

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Identify the conic represented by the polar equationr=31-2cosθ. Find the rectangular equation

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

Step 3. Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Statistics

Geometry

Logic and Functions

Mechanics Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Identify the conic represented by the polar equation $r = \frac{3}{1 - 2 \cos θ}$ . Find the rectangular equation