Question 411564
  <pre><font size = 3 color = "indigo"><b>
Hi
Ellipses:
1) 4x^2+5y^2-24x-20y+36=0
   4(x^2-6x) + 5(y^2-4y) +36 = 0   |completing squares
   4[(x-3)^2 -9] + 5[(y-2)^2 -4] + 36 = 0
   4(x-3)^2 -36 + 5(y-2)^2 -20 + 36 = 0
   4(x-3)^2 + 5(y-2)^2  = 20
  {{{ (x-3)^2/5 + (y-2)^2 /4 = 1 }}}  
C(3,2) V(3,4)&V(3,0) and V(3+sqrt(5),2)&V(3-sqrt(5),2)
    sqrt(5-4)= 1    f(4,2) and f(2,2)  
{{{drawing(300,300,  -6, 6, -6, 6,  arc(3,2,2sqrt(5),4),
 grid(1),
circle(3, 2,0.2),
graph( 300, 300, -6, 6, -6, 6 ))}}}

2) x^2+3y^2+2x-24y+46=0            |completing squares
   (x+1)^2 -1 +3[(y-4)^2 -16] + 46 = 0
   (x+1)^2 +3(y-4)^2 -49 + 46 = 0
   {{{(x+1)^2/3 + (y-4)^2/1 = 1}}}     
 C(-1,4) V(-1,5)&V(-1,3) and V(-1+sqrt(3),4)&V(-1-sqrt(3),4)
  sqrt(3-1)=sqrt(2)       f(-1 +sqrt(2),4) and f(-1-sqrt(2),4)  
{{{drawing(300,300, -6, 6, -6, 6,  arc(-1,4,2sqrt(3),2),
 grid(1),
circle(-1, 4,0.2),
graph( 300, 300, -6, 6, -6, 6))}}}