Question

Write the rectangular equation (x+5) 2 + y 2 = 25 in polar form.

Answer 1

To convert the given rectangular equation to polar form, we substitute x = r × cos(θ) and y = r × sin(θ) into the equation, simplify, and use trigonometric identities to arrive at the polar equation r = -10 × cos(θ).

Step-by-step explanation:

To convert the rectangular equation (x+5)^2 + y^2 = 25 to polar form, we will use the relationships between rectangular coordinates (x, y) and polar coordinates (r, θ). In polar coordinates, the x and y positions of a point are expressed as:

• x = r × cos(θ)
• y = r × sin(θ)

Now, let's substitute these expressions into the rectangular equation:

(r × cos(θ) + 5)^2 + (r × sin(θ))^2 = 25

Expanding the square and simplifying:

r^2 × cos^2(θ) + 10r × cos(θ) + 25 + r^2 × sin^2(θ) = 25

Using the Pythagorean identity cos^2(θ) + sin^2(θ) = 1:

r^2(1) + 10r × cos(θ) = 0

r^2 + 10r × cos(θ) = 0

This simplifies the polar equation:

r = -10 × cos(θ)

This is the polar form of the original rectangular equation.

Answer 2

r = -10*cos(t)

Step-by-step explanation:

To write the rectangular equation in polar form we need to replace x and y by:

`x=r*cos(t)\\y=r*sin(t)`

Replacing on the original equation, we get:

`(x+5)^2+y^2=25\\x^2+10x+25+y^2=25\\(r*cos(t))^2+10*(r*cos(t))+25+(r*sin(t))^2=25`

Using the identity
`sin^2(t)+cos^2(t)=1` and solving for r, we get that the polar form of the equation is:

`(r*cos(t))^2+10*(r*cos(t))+25+(r*sin(t))^2=25\\r^2cos^2(t)+10rcos(t)+r^2sin^2(t)=0\\r^2cos^2(t)+r^2sin^2(t)=-10rcos(t)\\r^2(cos^2(t)+sin^2(t))=-10rcos(t)\\r^2=-10rcos(t)\\\\r=-10cos(t)`