Respuesta :
The polar form of the parametric equation is option (D), r = 4 tan theta sec theta is the correct answer.
What is a polar from?
Polar form of a complex number is represented by a line whose length is the amplitude and by the phase angle. A polar form of a vector is denoted by ( , ) , where represents the distance from the origin and represents the angle measured from the -axis.
For the given situation,
The equation is x = 2t and y = t^2
Consider, [tex]x=2t\\[/tex]
⇒ [tex]t=\frac{x}{2}[/tex]
Now substitute t in y equation,
⇒ [tex]y=(\frac{x}{2})^{2}[/tex]
⇒ [tex]y=\frac{x^{2} }{4}[/tex]
We know that in polar form,
[tex]x=rcos\theta[/tex] and [tex]y=rsin \theta[/tex]
Now y becomes,
⇒ [tex]rsin\theta=\frac{(rcos\theta)^{2} }{4}[/tex]
⇒ [tex]4rsin\theta = r^{2}cos^{2}\theta[/tex]
⇒ [tex]r=4\frac{sin\theta }{cos^{2}\theta}[/tex]
we know that, [tex]\frac{sin\theta}{cos\theta} =tan\theta[/tex] and [tex]\frac{1}{cos\theta} = sec\theta[/tex]
⇒ [tex]r= 4tan\theta sec\theta[/tex]
Hence we can conclude that the polar form of the parametric equation is option (D), r = 4 tan theta sec theta is the correct answer.
Learn more about the polar form here
https://brainly.com/question/2272407
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