The Coefficient Scale Trap
The equation of a circle in the \(xy\)-plane is given by:
\(2x^2 + 2y^2 - 12x + 16y = 10\)
What is the area of the circle?
🚩 The Trap
Plunging directly into "completing the square" while the coefficients of \(x^2\) and \(y^2\) are still \(2\). If you don't divide by \(2\) first, your radius calculation will be off by a factor of \(\sqrt{2}\), making your area off by double!
✅ The Practix Shortcut
- The Mandatory Step: Standard form requires the coefficients of \(x^2\) and \(y^2\) to be \(1\). Divide the entire equation by \(2\): \[x^2 + y^2 - 6x + 8y = 5\]
- The Center Shortcut: Half the coefficients of \(x\) and \(y\), then flip the signs: \[h = -(-6)/2 = 3, \quad k = -(8)/2 = -4\]
- The Radius Formula: \(r^2 = h^2 + k^2 + \text{constant term from RHS}\): \[r^2 = 3^2 + (-4)^2 + 5 = 9 + 16 + 5 = 30\]
- Calculate Area: Area \(= \pi r^2 = 30\pi\).
Answer: \(30\pi\).