Find a cartesian equation for the curve and identify it. r = 3 tan(θ) sec(θ)

Solution:

Given, r = 3 tan(θ) sec(θ)

We have to find a cartesian equation for the curve.

For a cartesian equation,

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 = 3 tan (θ) sec (θ)

r = 3 (y/x) (r/x)

r = 3yr/x²

On simplification,

r(x²) = 3yr

x² = 3y

y = (1/3)x²

So the curve is a parabola opening upwards with vertex (0,0).

Therefore, the cartesian equation is y = (1/3)x².

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Find a cartesian equation for the curve and identify it. r = 3 tan(θ) sec(θ)

Summary:

A cartesian equation for the curve r = 3 tan (θ) sec (θ) is y = (1/3)x².