Question

$x^2+y^2-8x+4y-5=0$the variables r and u represent polar coordinates. (a) Write each polar equation as an equation in rectangular coordinates (x, y). (b) Identify the equation and graph it.

$r^2+4r\sin \theta -8r\cos \theta =5$

Short answer

Polar equation as an equation in rectangular coordinates is $x^2+y^2-8x+4y-5=0$

Step-by-step solution

Step 1. Given data

$r2+4r\sin u-8r\cos u=5$

Step 2. Substitute the values of sinθ and cosθ and simplify the equation

we know,

$\sin \theta =\frac{y}{\sqrt{x^2+y^2}}$and $\cos \theta =\frac{x}{\sqrt{x^2+y^2}}$

Substituting in the given equation,

$x^2+y^2+4\left(\sqrt{x^2+y^2}\right)\left(\frac{y}{\sqrt{x^2+y^2}}\right)-8\left(\sqrt{x^2+y^2}\right)\left(\frac{x}{\sqrt{x^2+y^2}}\right)=5$

Above equation can be simplified as,$x^2+y^2-8x+4y-5=0$

Step 3. Plotting the graph