Eliminate the parameter to find the cartesian equation of the curve
$x = 5 \sec θ, y = 5 \tan θ, - \frac{π}{2} < θ < \frac{π}{2} .$

The equation of the curve is:
$x =$

Answer

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Answer

The cartesian equation of the curve is $x^{2} = 25 + y^{2}$ .

Steps

Step 1 ：First, we express $\sec θ$ and $\tan θ$ in terms of $x$ and $y$ respectively.

Step 2 ：From $x = 5 \sec θ$ , we get $\sec θ = \frac{x}{5}$ .

Step 3 ：From $y = 5 \tan θ$ , we get $\tan θ = \frac{y}{5}$ .

Step 4 ：Now, we substitute these expressions into the identity $\sec^{2} θ = 1 + \tan^{2} θ$ to get an equation in terms of $x$ and $y$ only.

Step 5 ：This gives us ${(\frac{x}{5})}^{2} = 1 + {(\frac{y}{5})}^{2}$ .

Step 6 ：Simplifying this equation gives us the cartesian equation of the curve.

Step 7 ：The cartesian equation of the curve is $x^{2} = 25 + y^{2}$ .