¿HIPERBOLAS?
Para cada una de las siguientes ecuaciones, reducir a su forma ordinaria y hallar : a) las coordenadas del centro , b) las longitudes de los ejes transverso, conjugado y principal c) las coordenadas de los vértices y focos, d) la excentricidad, la longitud del lado recto y e) la ecuación de sus asíntotas. Trazar el lugar geométrico.
1. x2 - 9y2 - 4x + 36y - 41 =0
2. 4x2 - 9y2 + 32x + 64 =0
3. x2 - 4y2 - 2x+1=0
4. 9x2 - 4y2 + 54x + 16y + 29 =0
5. 3x2 - y2 + 30x + 78 =0
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Pulgar arriba y tu comentario.
x² - 9y² - 4x + 36y - 41 = 0
x² - 4x - 9y² + 36y = 41
[x² - 4x] - [9y² - 36y] = 41
[x² - 4x] - 9.[y² - 4y] = 41
[x² - 4x + (4 - 4)] - 9.[y² - 4y + (4 - 4)] = 41
[x² - 4x + 4 - 4] - 9.[y² - 4y + 4 - 4] = 41
[x² - 4x + 4] - 4 - 9.[y² - 4y + 4] + 36 = 41
[x² - 4x + 4] - 9.[y² - 4y + 4] = 9
(x - 2)² - 9.(y - 2)² = 9
[(x - 2)²/9] - (y - 2)² = 1
[(x - 2)²/3²] - (y - 2)² = 1 ← this is an hyperbola
4x² - 9y² + 32x + 64 = 0
4x² + 32x - 9y² = - 64
4.[x² + 8x] - 9y² = - 64
4.[x² + 8x + (16 - 16)] - 9y² = - 64
4.[x² + 8x + 16 - 16] - 9y² = - 64
4.[x² + 8x + 16] - 64 - 9y² = - 64
4.[x² + 8x + 16] - 9y² = 0
4.(x + 4)² - 9y² = 0
9y² = 4.(x + 4)²
(3y)² = [± 2.(x + 4)]²
3y = ± 2.(x + 4)
y = ± 2.(x + 4)/3
y₁ = 2.(x + 4)/3 ← this is a straight line
y₂ = - 2.(x + 4)/3 ← this is another straight line
x² - 4y² - 2x + 1 = 0
x² - 2x + 1 - 4y² = 0
(x - 1)² - 4y² = 0
(x - 1)² - (2y)² = 0 → you recognize: a² - b² = (a + b).(a - b)
[(x - 1) + 2y].[(x - 1) - 2y] = 0
(x - 1 + 2y).(x - 1 - 2y) = 0
First case: (x - 1 + 2y) = 0
2y = - x + 1
y = (- x + 1)/2
y₁ = - (1/2).x + (1/2) ← this is a straight line
Second case: (x - 1 - 2y) = 0
2y = x - 1
y = (x - 1)/2
y₂ = (1/2).x - (1/2) ← this is another straight line
9x² - 4y² + 54x + 16y + 29 = 0
9x² + 54x - 4y² + 16y = - 29
9.[x² + 6x] - 4.[y² - 4y] = - 29
9.[x² + 6x + (9 - 9)] - 4.[y² - 4y + (4 - 4)] = - 29
9.[x² + 6x + 9 - 9] - 4.[y² - 4y + 4 - 4] = - 29
9.[x² + 6x + 9] - 81 - 4.[y² - 4y + 4] + 16 = - 29
9.[x² + 6x + 9] - 4.[y² - 4y + 4] = 36
9.(x + 3)² - 4.(y - 2)² = 36
[9.(x + 3)²/36] - [4.(y - 2)²/36] = 1
[(x + 3)²/4] - [(y - 2)²/9] = 1
[(x + 3)/2]² - [(y - 2)/3]² = 1 ← this is an hyperbola
3x² - y² + 30x + 78 = 0
3x² + 30x - y² = - 78
3.[x² + 10x] - y² = - 78
3.[x² + 10x + (25 - 25)] - y² = - 78
3.[x² + 10x + 25 - 25] - y² = - 78
3.[x² + 10x + 25] - 75 - y² = - 78
3.[x² + 10x + 25] - y² = - 3
3.(x + 5)² - y² = - 3
(x + 5)² - (y²/3) = - 1 ← this is an hyperbola
How to calculate the different coefficients? This following link:
https://en.wikipedia.org/wiki/Hyperbola