Please check:
1.Find polar equation of x^2 + y^2 = 10y
r^2 for x^2 + y^2 and sin(theta sin for y
=r^2=10sin(theta)
2. y=5
r=5/sin
3. x^2+y^2
=x^2 + y^2 =2x+3y
To find the polar equation of the given equations, we'll need to convert them into polar form by substituting the Cartesian coordinates with polar coordinates. Let's break down each equation step by step:
1. Cartesian equation: x^2 + y^2 = 10y
To convert this equation into polar form, we'll use the following substitutions:
- x = r * cos(theta)
- y = r * sin(theta)
Substituting these values into the equation, we get:
(r * cos(theta))^2 + (r * sin(theta))^2 = 10 * (r * sin(theta))
Simplifying the equation:
r^2 * cos^2(theta) + r^2 * sin^2(theta) = 10r * sin(theta)
Simplifying further:
r^2 * (cos^2(theta) + sin^2(theta)) = 10r * sin(theta)
Since cos^2(theta) + sin^2(theta) is equal to 1, we have:
r^2 = 10r * sin(theta)
Therefore, the polar equation for x^2 + y^2 = 10y is r^2 = 10r * sin(theta).
2. Cartesian equation: y = 5
To convert this equation into polar form, we'll use the same substitutions:
- x = r * cos(theta)
- y = r * sin(theta)
Substituting the value of y, we get:
r * sin(theta) = 5
Therefore, the polar equation for y = 5 is r * sin(theta) = 5.
3. Cartesian equation: x^2 + y^2 = 2x + 3y
This equation does not directly convert into polar form. However, we can rewrite it by completing the square for the x terms and y terms separately.
Let's rearrange the equation:
x^2 - 2x + y^2 - 3y = 0
Completing the square for x terms:
(x^2 - 2x + 1) + y^2 - 3y = 1
(x - 1)^2 + y^2 - 3y = 1
Completing the square for y terms:
(x - 1)^2 + (y^2 - 3y + 9/4) = 1 + 9/4
(x - 1)^2 + (y - 3/2)^2 = 13/4
Now, we can see that the equation is in the standard form:
r^2 = (x - h)^2 + (y - k)^2
So, the polar equation for x^2 + y^2 = 2x + 3y is:
r^2 = (x - 1)^2 + (y - 3/2)^2 = 13/4.
I hope this explanation helps you understand how to convert Cartesian equations to polar form. Let me know if you have any more questions!