SOLUTION: find the centre,foci,vertices,and graph the following conics 1)8y²+12x=0 2)9x²-4y²-18x-16y-43=0 3)9x²+4y²-18x+16y-11=0

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: find the centre,foci,vertices,and graph the following conics 1)8y²+12x=0 2)9x²-4y²-18x-16y-43=0 3)9x²+4y²-18x+16y-11=0      Log On


   

Question 1127410: find the centre,foci,vertices,and graph the following conics
1)8y²+12x=0
2)9x²-4y²-18x-16y-43=0
3)9x²+4y²-18x+16y-11=0
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
find the center,foci,vertices,and graph the following conics
1)
8y%5E2%2B12x=0 -> Parabola with horizontal axis:%28y-k%29%5E2=4p%28x-h%29
Vertex is (h,k)
Focus is (h%2Bp,k)
Directrix is the line x=h-p
Axis is the line y=k

8y%5E2=+-12x
y%5E2=-%2812%2F8%29x
y%5E2=-%283%2F2%29x...i you compare to %28y-k%29%5E2=4p%28x-h%29, you see that
=> h=0, k=0 ->Vertex is (0,0)
Axis is the line y=0
4p=-%283%2F2%29 -> p=-%283%2F%282%2A4%29%29 -> p=-%283%2F8%29
Focus is (h%2Bp,k)->(0-3%2F8,0)->(-3%2F8,0)
Directrix is the line x=h-p ->x=0-%28-3%2F8%29->+x=3%2F8



2)
9x%5E2-4y%5E2-18x-16y-43=0 ->hyperbola
9x%5E2-18x-4y%5E2-16y=43
%289x%5E2-18x%29-%284y%5E2%2B16y%29=43
9%28x%5E2-2x%29-4%28y%5E2%2B4y%29=43
9%28x%5E2-2x%2Bb%5E2%29+-9b%5E2+-4%28y%5E2%2B4y%2Bb%5E2%29+-4b%5E2=43
9%28x%5E2-2x%2Bb%5E2%29+-9b%5E2+-4%28y%5E2%2B4y%2Bb%5E2%29+-4b%5E2=43
since a=1,2ab=2->b=1; for y part a=1, 2ab=4->b=2
9%28x-1%29%5E2+-9%2A1%5E2+-4%28y%2B2%29%5E2+-%28-4%29%2A2%5E2=43
9%28x-1%29%5E2+-9+-4%28y%2B2%29%5E2+%2B16=43
9%28x-1%29%5E2++-4%28y%2B2%29%5E2+%2B7=43
9%28x-1%29%5E2++-4%28y%2B2%29%5E2+=43-7
9+%28x+-+1%29%5E2+-+4+%28y+%2B+2%29%5E2++=+36
9+%28x+-+1%29%5E2+%2F36-+4+%28y+%2B+2%29%5E2%2F36++=+36%2F36
%28x+-+1%29%5E2+%2F4-++%28y+%2B+2%29%5E2%2F9++=+1
Center is (h,k) .
c%5E2=a%5E2%2Bb%5E2
h=1
k=-2
a=2 ->semi-major axis length
b=3->semi-minor axis length
c=sqrt%282%5E2%2B3%5E2%29=sqrt%2813%29
center: (1,-2)
vertices: (-1,+-2) and (3,+-2)
foci: (1+-+sqrt%2813%29, -2) and (1+%2B+sqrt%2813%29, -2) ≈(-2.6, -2) and (4.6, -2)




3)
9x%5E2%2B4y%5E2-18x%2B16y-11=0 ->ellipse
%289x%5E2-18x%29%2B%284y%5E2%2B16y%29=11
9%28x%5E2-2x%29%2B4%28y%5E2%2B4y%29=11
9%28x%5E2-2x%2B1%5E2%29+-9%2A1%5E2%2B4%28y%5E2%2B4y%2B2%5E2%29+-4%2A2%5E2=11
9%28x-1%29%5E2+-9%2B4%28y%2B2%29%5E2-16=11
9%28x-1%29%5E2+%2B4%28y%2B2%29%5E2-25=11
9%28x-1%29%5E2+%2B4%28y%2B2%29%5E2=11%2B25
9%28x-1%29%5E2+%2B4%28y%2B2%29%5E2=36
9%28x-1%29%5E2%2F36+%2B4%28y%2B2%29%5E2%2F36=36%2F36
%28x-1%29%5E2%2F4+%2B%28y%2B2%29%5E2%2F9=1->Ellipse with vertical major axis:%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2=1

h=1
k=-2
a=3 ->semi-major axis length
b=2->semi-minor axis length
c=sqrt%282%5E2%2B3%5E2%29=sqrt%2813%29
center: (1,-2)
foci : (1, -2+-+sqrt%285%29) and (1, -2+%2B+sqrt%285%29)≈(1, -4.2) and (1,+0.2)
vertices: (1,+-5) and (1, 1)