Question

Show that$A+C=A\text{'}+C\text{'}$, and thus show that $A+C$is invariant; that is, its value does not change under a rotation of axes.

Short answer

Thus,$A+C=A\text{'}+C\text{'}$is invariant.

Step-by-step solution

Step 1. Given Information

$A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$

Step 2. Explanation to invariant

Consider the equation $x^2+4xy+4y^2+5\sqrt{5}y+5=0$

Here $A=1,C=4$gives $A+C=1+4=5$

The rotated equation is $0x{\text{'}}^2+5y{\text{'}}^2-2.3\sqrt{5}x\text{'}+4.4\sqrt{5}y\text{'}+5=0$

Here $A\text{'}=0,C\text{'}=5$gives $A\text{'}+C\text{'}=5$

Both$A+C$and$A\text{'}+C\text{'}$are equal.