How do you convert r=2sec^2(theta/2) into cartesian form?

1 Answer
Apr 20, 2018

The Cartesian equation is y^2=-8(x-2).

Explanation:

To convert the equation, use the reciprocal definition:

sectheta=1/costheta

and the cosine half-angle formula:

cos(theta/2)=+-sqrt((1+costheta)/2)

Here's the equation:

r=2sec^2(theta/2)

r=2*1/cos^2(theta/2)

r=2*1/(cos(theta/2))^2

r=2*1/(+-sqrt((1+costheta)/2))^2

r=2*1/((1+costheta)/2)

r=4/(1+costheta)

r+rcostheta=4

Using the substitutions r=sqrt(x^2+y^2) and rcostheta=x:

sqrt(x^2+y^2)+x=4

sqrt(x^2+y^2)=4-x

x^2+y^2=(4-x)^2

color(red)cancelcolor(black)(x^2)+y^2=16-8x+color(red)cancelcolor(black)(x^2)

y^2=-8x+16

y^2=-8(x-2)

That's the equation; it's leftward-opening parabola. Here's the graph:

graph{y^2=-8(x-2) [-25.66, 25.65, -12.83, 12.83]}

Hope this helped!