y^2/25-x^2/9=0
What is the polar equation that is equivalent to the rectangular equation?
Sorry it's actually equal to 1 not 0
just make the usual substitution.
y = rsinθ
x = rcosθ
so, that means
r^2sin^2θ/25 - r^2cos^2θ/9 = 1
r^2(9sin^2θ-25cos^2θ) = 225
r^2 = 225/(9sin^2θ-25cos^2θ)
see the expected vertical hyperbola at
http://www.wolframalpha.com/input/?i=plot+r^2+%3D+225%2F%289sin^2%CE%B8-25cos^2%CE%B8%29
To convert the given rectangular equation to a polar equation, we can use the following conversion equations:
x = r × cosθ
y = r × sinθ
Substituting these conversion equations into the given rectangular equation:
(y^2 / 25) - (x^2 / 9) = 0
We get:
[(r × sinθ)^2 / 25] - [(r × cosθ)^2 / 9] = 0
Simplifying further:
(r^2 × sin^2θ) / 25 - (r^2 × cos^2θ) / 9 = 0
Multiplying both sides by 225:
9r^2 × sin^2θ - 25r^2 × cos^2θ = 0
Now, factoring out r^2:
r^2 × (9sin^2θ - 25cos^2θ) = 0
Since this should be true for any value of r, for the equation to hold true, the factor (9sin^2θ - 25cos^2θ) must equal zero:
9sin^2θ - 25cos^2θ = 0
Therefore, the polar equation that is equivalent to the given rectangular equation is:
r^2 = 25cos^2θ / 9 - sin^2θ
To obtain the polar equation that is equivalent to the given rectangular equation, we can use the following conversion formulas:
x = r * cos(theta)
y = r * sin(theta)
First, let's rewrite the given equation:
(y^2 / 25) - (x^2 / 9) = 0
Since we want the polar equation, we need to express x and y in terms of r and theta. Rearrange the equation by multiplying both sides by 225 to eliminate the denominators:
225 * ((y^2 / 25) - (x^2 / 9)) = 0
This simplifies to:
9y^2 - 25x^2 = 0
Now, substitute x = r * cos(theta) and y = r * sin(theta) into this equation:
9(r * sin(theta))^2 - 25(r * cos(theta))^2 = 0
Simplifying further:
9r^2 * sin^2(theta) - 25r^2 * cos^2(theta) = 0
Divide through by r^2:
9sin^2(theta) - 25cos^2(theta) = 0
Now, we can rewrite this equation in terms of the polar variables r and theta:
9sin^2(theta) - 25cos^2(theta) = 0
This is the polar equation that is equivalent to the given rectangular equation.