Find the the length in the first quadrant of the circle described by the polar equation r = 2 sin t + 4 cos t

Answer 1

The equation of the circle is given in polar form. We will convert it to Cartesian.

r = 2 sin t + 4 cos t
r = 2y/r + 4x/r
r² = 2y + 4x
x² + y² = 2y + 4x
x² - 4x + y² - 2y = 0
(x² - 4x + 4) + (y² - 2y + 1) = 4 + 1
(x - 2)² + (y - 1)² = 5

So this is an equation of a circle with center (2,1) and radius √5.

The x intercepts are (4,0) and (0,0).
The y intercepts are (0,2) and (0,0).

The line thru the x and y intercepts (4,0) and (0,2) also passes thru the center of the circle (2,1) which means it is a diameter. On one side of the diameter the circle is in the first quadrant. On the other side of the diameter the circle is in the second or fourth quadrant. It does touch the origin at one point but never enters the first quadrant.

So exactly half the circle is in the first quadrant. That would be half of the circumference or

πr = π√5

Answer 2

I presume you want to find the arc length of the circle
so now first we have to concert this from a polar for to a parametric form
x = r*cos(ang)
y = r* cos(ang)
so
x = (2sin t + 4cos t) (cos(ang))
y = (2sing t +4cos t)(sin(ang))
no both t and ang are the same - angles in radian measure
s i will jus us t for simplicity

now to find an arc length using parametric equation u need this formula
b
l = S(((dx/dt)^2 + (dy/dt)^2))^(.5))
a
S is the intergral sign
if this is confusing pls refer to you text book for the furmula for arc length
now we find
x' and y'
and since this in the 1st quadrant we want the limts to be in the 1st quadrant, use a graphing calculator to see that. NOw plug all in and vola you got the answer
email at [e-mail] if u need clearlr

Answer 3

Hi

Not circle

Ellipse

Length : Elliptic integral :)
Only numerical resoluble

bye