Find a cartesian equation for the curve and identify it. r = 2 tan(θ) sec(θ)

Solution:

Given, r = 2 tan(θ) sec(θ)

We have to find a cartesian equation for the curve.

For a cartesian equation, the cartesian coordinates are (x,y)

x = r cos θ

y = r sin θ

So, cos θ = x/r

sin θ = y/r

Now, tan θ = sin θ/cos θ

= (y/r) / (x/r)

tan θ = y/x

sec θ = 1/cos θ

= 1/(x/r)

sec θ = r/x

Now, r = 2 tan (θ) sec (θ)

r = 2 (y/x) (r/x)

r = 2yr/x²

On simplification,

r(x²) = 2yr

x² = 2y

y = (1/2)x²

This equation is that of a parabola.

Therefore, the cartesian equation for the curve is y = (1/2)x².

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Find a cartesian equation for the curve and identify it. r = 2 tan(θ) sec(θ)

Summary:

A cartesian equation for the curve r = 2 tan (θ) sec (θ) is y = (1/2)x².