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# Find a cartesian equation for the curve and identify it. r = 6 tan(θ) sec(θ)

**Solution:**

Given, r = 6 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 = 6 tan (θ) sec (θ)

r = 6 (y/x) (r/x)

r = 6yr/x^{2}

On simplification,

r(x^{2}) = 6yr

x^{2} = 6y

y = (1/6)x^{2}

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

Therefore, the cartesian equation is y = (1/6)x^{2}.

## Find a cartesian equation for the curve and identify it. r = 6 tan(θ) sec(θ)

**Summary:**

A cartesian equation for the curve r = 6 tan (θ) sec (θ) is y = (1/6)x^{2}.

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