Find a cartesian equation for the curve and identify it. r = 7 tan(θ) sec(θ)

Solution:

Given,

r = 7 tan(θ) sec(θ)

For a cartesian equation,

x = r cos θ

y = r sin θ

cos θ = x/r; sin θ = y/r

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

r = 7 tan(θ) sec(θ)

r = 7 (y/x)(r/x)

r = 7yr/x²

x²r = 7yr

x² = 7y

y = (1/7)x²

The curve is a parabola opening upwards with vertex (0, 0)

Therefore, the cartesian equation is y = (1/7)x².

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Find a cartesian equation for the curve and identify it. r = 7 tan(θ) sec(θ)

Summary:

A cartesian equation for the curve r = 7 tan(θ) sec(θ) is y = (1/7)x².