Question

# A polar conic section Consider the equation r^2 = sec 2 theta. Convert the equation to Cartesian coordinates and identify the curve

Conic sections
A polar conic section Consider the equation $$r^2 = sec 2 \theta$$. Convert the equation to Cartesian coordinates and identify the curve

Given: we have an equation in polar form $$r^2 = \sec 2 \theta$$ Part a) to determinate the equation to Cartesian coordinates and identify the curve Explain: to convert the equation we can write the equation sa follows $$r = \sqrt{\sec 2 \theta}$$
$$r = {1}{\sqrt{\cos 2 \theta}}$$ We know these relations $$x = r \cos \theta, y = r \sin \theta, r = \sqrt{x^2 + y^2}$$ Can be written $$\cos \theta$$ $$= x/r, \sin \theta = y/r$$ $$r = \frac{1}{\sqrt{\cos^{2} \theta}} = \frac{1}{\sqrt{\cos^2 \theta - \sin^2 \theta}} [\cos^{2} \theta = \cos^2 \theta - \sin^2 \theta]$$
Putting in the equation we have $$r = \frac{1}{\sqrt{\frac{x}{r}}}^2 - \frac{y}{r}^2$$ $$\frac{x^2}{r^2} - \frac{y^2}{r^2} = \frac{1}{r^2}$$ We have the equation $$x^2 - y^2 = 1$$