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