Math Ellipse Worksheet Solution Key

Pre-Calc Worksheet Ellipses Name Key Ellipses Worksheet Graph the ellipse and identify the center, vertices, and foci. (\frac{x^{2}}{16}+\frac{y^{2}}{4}=1) a=4 \(b=2\) 2. \(\frac{x^{2}}{9}+\frac{y^{2}}{36}=1\) \[C^{2}=16-4=12\] \[a=6\] \[c=2\sqrt{3}\] \[b=3\] \[c^{2}=36-9=27\] \[c=\sqrt[3]{3}\] Center: (0,0) Center: (0,0) Vert: (4,0);(-4,0) Vert: (0,6); (0,-6) cv: (0,2); (0,-2) CV: (3,0); (-3,0) Foci: \((2\sqrt{3},0);(-2\sqrt{3},0)\) Foci: \((0,3\sqrt{3});\) (0,-313 3. 25x2+4y² 100 4. \(7x^{2}=35-5y^{2}\) \[\frac{x^{2}}{4}+\frac{y^{2}}{25}=1\] \[7x^{2}+5y^{2}=35\] \[a=5\] \[\frac{x^{2}}{5}+\frac{y^{2}}{7}=1\] \[b=2\] \[a=\sqrt{7}\] \[c^{2}=25-4=21\] \[b=\sqrt{5}\] \[c=\pm\sqrt{21}\] \[c^{2}=7-5=2\] \[C=\pm\sqrt{2}\] Center. (0,0) Center: (0,0) → Vert (0,5); (0,-5) \[lert.(0,\sqrt{7});(0,-\sqrt{7})\] \[\exists V:(2,0);(-2,0)\] cv: \((\sqrt{5},0);(-\sqrt{5},0)\) \[oct(0,\sqrt{2}1);(0,-\sqrt{21})\] Foci \((0,\sqrt{2});(0,-\sqrt{2})\) \[\frac{(x-4)^{2}}{9}+\frac{(y+2)^{2}}{25}=1\] [\frac{(x-2)^{2}}{9}+\frac{(y-1)^{2}}{4}=1] [a=5] center ((2,1)) [b=3] [a=3] [c=4] [b=2] [c=\sqrt{5}] Center: (2,1) Center: (4,-2) Vert: (5,1); (-1,1) Vert: (4,3); (4,-7) CV: (2,3); (2,-1) \[\exists V:(1,-2);(7,-2)\] Foci \((2\pm\sqrt{5},1)\) Foci: (4,2); (4,-6) 7. (x+3) (y-2) 8. \[(x+3)^{2}+4(y-2)^{2}=16\] \[\frac{x^{2}}{25}+\frac{(y-2)^{2}}{36}=1\] \[a=4\] \[a=6\] \[b=2\] \[b=5\] \[C=2\sqrt{3}\] \[c=\sqrt{11}\] + 41 Center: (-3,2) Center: (0,2) Vert: (1,2);(-7,2) Vert: (0,8); (0,-4) cv: (-3,4); (-3,0) cv: (5,2); (-5,2) Foci: (-3±23, 2) Foci: \((0,1\pm\sqrt{n})\) Find the standard form of the equation of each ellipse. 10. 11. \[a=4\] \[a=2\] \[a=3\] \[b=3\] \[b=1\] b1 horiz Vert horiz cent (1,1) \[\frac{x^{2}}{9}+\frac{y^{2}}{1}=1\] \[\frac{x^{2}}{9}+\frac{y^{2}}{16}=1\] \[\frac{(x+1)^{2}}{4}+\frac{(y-1)^{2}}{1}=1\] Find the standard form of the equation of each ellipse satisfying the given conditions. Foci: (±5, 0); Vertices (±8, 0) c=5 a=8 13. Foci: (0, ±4); Vertices: \((0,\pm7) a=7 c=4\) cent (0,0) \[\frac{x^{2}}{64}+\frac{y^{2}}{3R}=1\] cent (0,0) \[\frac{x^{2}}{33}+\frac{y^{2}}{49}=1\] horiz \[25=64-b^{2}\] \[b^{2}=39\] vert \[16=49-b^{2}\] \[b^{2}=33\] 14. 15. Foci: (±2, 0); y-intercepts: ±3 Major axis horizontal with length 8; length of \[4=a^{2}-9\] minor axis 4; Center (0, 0) \(a=4 b=2\) \[c=2\] \[b=3\] 0,0) \[a^{2}=13\] \[\frac{x^{2}}{16}+\frac{y^{2}}{4}=1\] horiz. \[\frac{x^{2}}{1^{3}}+\frac{y^{2}}{9}=1\] \[2a=6\] \[a=3\] Vert! 16. Major axis vertical with length 10: 17. Length of minor axis 4; Center (-2, 3) (79) \[\underline{(x+2)^{2}}+\underline{(y-3)}^{2}=1\] Endpoints of Major Axis: (7, 9) & (7,3) Endpoints of Minor Axis (5, 6) & (9,6) \[2b=4\] \[b=2\] \[a=5\] \[b=2\] center \(:(7,6)\) \[\frac{(x-7)^{2}}{4}+\frac{(y-6)^{2}}{9}=1\] 4 25 Convert each equation to standard form by completing the square. 18. \[9x^{2}+25y^{2}-36x+50y-164=0\] 19. \[x^{2}+4y^{2}+10x-8y+13=0\] \[x^{2}+10x+4y^{2}-8y\] \[x-13\] \[x^{2}+bx+25+4(y^{2}-2y+1)=-13+25+4\] \[(x+5)^{2}+4(y-1)^{2}=16\] \[\frac{(x+5)}{16}+\frac{(y-1)}{4}=1\] \[9x^{2}-36x+25y^{2}+50y=164\] \[1(x^{2}-4x)+25(y^{2}+2y)=164\] \[9(x^{2}-4x+4)+25(y^{2}+2y+1)=164+31a+25}{16}\] \[9(x-2)^{2}+25(y+1)^{2}=225\] \[\frac{(x-2)^{2}}{25}+\frac{(y+1)^{2}}{9}=1\] [4x^{2}+16x] [4x^{2}+y^{2}+16x-6y-39=0] [+y^{2}-6y=39] [4(x^{2}+9x+4)+y^{2}-6y+9=89+16+9}] [4(x+2)^{2}+(y-3)^{2}=64] 21. [\frac{(x+2)^{2}}{11}+\frac{(y-3)^{2}}{11}=1] [4x^{2}+25y^{2}-24x+100y+36=0] [A(x^{2}-6x+9)+25(y^{2}+4y+4)=-36+36+100] [4(x-3)^{2}+25(y+2)^{2}=100] [(y-3)^{2},(y+2)^{2}]