R=8cos (theta) - 10sin (theta) to cartesian coordinates

Convert polar equation (R, Ф):

(1) R = 8cos Ф - 10sin Ф into Cartesian (x, y)

We know that:

x = R. cos Ф → cos Ф = x/R, and

y = R. sin Ф → sin Ф = y/R

and that x² + y² = R²

Replace in

R = 8cos Ф - 10sin Ф, cos Ф and sin Ф by the related x and y:

R = 8 (x/R) - 10 (y/R)

Multiply both sides by R:

R² = 8R (x/R) - 10R (y/r) ↔ R² = 8x - 10y, but we have also R² = x² + y². hence: 8x - 10y = x² + y²

OR x² + y² - 8x + 10y = 0

We can continue in completing th squares of (x² - 8x + ?) and (y² + 10y + ?)

(x² - 8x + ?) = (x - 4) ² - 16

and (y² + 10y + ?) = (y + 5) ² - 25

Final Equation:

(x - 4) ² - 16 + (y + 5) ² - 25 →→→ (x - 4) ² + (y + 5) ² = 41