Question

4x2 +9y2 = 72

x - y2 = -1
Select all of the following that are solutions to the system shown.
Solve?

Answer 1

`\bold{\left((3)/(13)\ ,\ (4√(13))/(13)\right)\qquad\vee\qquad \left((3)/(13)\ ,\ -(4√(13))/(13)\right)}`

Explanation:

x - y² = -1 ⇒ - y² = - x - 1 ⇒ y² = x + 1 {x≥-1}

4x² + 9y² = 72

4x² + 9(x + 1)² = 72

4x² + 9(x² + 2x + 1) = 72

4x² + 9x² + 18x + 9 - 72 = 0

13x² + 18x - 63 = 0 ⇒ a = 13, b = 18, c = -63

`x=(-18\pm√(18^2-4\cdot13\cdot(-63)))/(2\cdot13)=(-18\pm√(324+252))/(26)=(-18\pm√(576))/(26)\\\\x=(-18+24)/(26)=\frac3{13}\qquad\vee\qquad x=(-18-24)/(26)=-(21)/(13)\\\\ y^2= x + 1\quad\iff\quad x\ge-1\\\\y^2=\frac3{13}+1\\\\ y^2=(16)/(13)\\\\y=(4)/(√(13))=(4√(13))/(13)\qquad\vee\qquad y=-(4√(13))/(13)`