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Question 1181120:
Found 2 solutions by ikleyn, MathLover1:
Answer by ikleyn(52799) About Me  (Show Source):
You can put this solution on YOUR website!
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D U P L I C A T E


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                Q U A D R U P L I C A T E

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
9x%5E2+%2B+16y%5E2+-54x+-+64y+%2B+1+=+0
%289x%5E2-54x%29+%2B+%2816y%5E2++-+64y%29+=-1
9%28x%5E2-6x%29+%2B16+%28y%5E2++-+4y%29+=-1
9%28x%5E2-6x%2Bb%5E2%29+-9b%5E2%2B16+%28y%5E2++-+4y%2Bb%5E2%29+-16b%5E2=-1
9%28x%5E2-6x%2B3%5E2%29+-9%2A3%5E2%2B16+%28y%5E2++-+4y%2B2%5E2%29+-16%2A2%5E2=-1
9%28x-3%29%5E2+-81%2B16+%28y+-+2%29%5E2+-64=-1
9%28x-3%29%5E2+%2B16+%28y+-+2%29%5E2+-145=-1
9%28x-3%29%5E2+%2B16+%28y+-+2%29%5E2+=145-1
9%28x-3%29%5E2+%2B16+%28y+-+2%29%5E2+=144..........both sides divide by 144
9%28x-3%29%5E2%2F144+%2B16+%28y+-+2%29%5E2%2F144+=144%2F144
%28x-3%29%5E2%2F16+%2B+%28y+-+2%29%5E2%2F9+=1
=> h=3, k=2,
a=sqrt%2816%29=>a=4
b=sqrt%289%29=>b=3
a%3Ebtherefore a is semi-major axis and b is semi-minor axis
c=sqrt%28a%5E2-b%5E2%29=>c=sqrt%284%5E2-3%5E2%29=>c=sqrt%287%29
1. Center: (3,2)
2. Foci: (h%2Bc, k ), (h-c, k )
𝐹1 (3%2Bsqrt%287%29, 2 ), (3-sqrt%287%29, 2 )
𝐹2 (5.7, 2 ), (-0.4, 2 )


3. Vertices: (h%2Ba,k), (h-a,k)
𝑉1 (3%2B4,2) => (7,2)
𝑉2 (3-4,2) => (-1,2)
4. Co-vertices (h,k%2Bb), (h,k-b)


𝐵1 (3,2%2B3) =>(3,5)
𝐵2(3,2-3)=>(3,-1)
5. Endpoints of Latus Rectum
foci are (5.7, 2 ), (-0.4, 2 )
endpoints will lie on intersection of the line x=5.7 and the line x=-0.4 with ellipse
%28x-3%29%5E2%2F16+%2B+%28y+-+2%29%5E2%2F9+=1...substitute x
%285.7-3%29%5E2%2F16+%2B+%28y+-+2%29%5E2%2F9+=1...solve for y
0.455625%2B+%28y+-+2%29%5E2%2F9+=1
+%28y+-+2%29%5E2+=9%281-0.455625%29
+%28y+-+2%29%5E2+=4.899375
+y++=sqrt%284.899375%29%2B2
+y++=4.2 or +y=-0.2
%28x-3%29%5E2%2F16+%2B+%28y+-+2%29%5E2%2F9+=1...substitute x
%28-0.4-3%29%5E2%2F16+%2B+%28y+-+2%29%5E2%2F9+=1...solve for y
0.7225%2B+%28y+-+2%29%5E2%2F9+=1
+%28y+-+2%29%5E2+=9%281-0.7225%29
+%28y+-+2%29%5E2+=2.4975
+y++=sqrt%282.4975%29%2B2
+y++=3.6 or +y++=0.4

𝐸1 (5.7, 4.2 )
𝐸2 (5.7, -0.2 )
𝐸3 (-0.4, 3.6 )
𝐸4 (-0.4, 0.4 )

6. Directrices:
x=3-16sqrt%287%29%2F7=>approximately x=-3.1
x=3%2B16sqrt%287%29%2F7=>approximately x=9.1

7. Eccentricity: sqrt%287%29%2F4 approximately 0.7
8. Length of LR:9%2F2
9. Length of Major Axis:4
10.Length of Minor Axis:3