Answer :
To determine which of the given equations represent hyperbolas, we need to analyze the general form of a conic section equation [tex]\( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \)[/tex]. For a conic section to be a hyperbola, the discriminant [tex]\( \Delta = B^2 - 4AC \)[/tex] must be greater than 0 (i.e., [tex]\( \Delta > 0 \)[/tex]).
Here, let's review the given equations one by one:
1. [tex]\( 49x^2 - 98x - 64y^2 + 256y - 2831 = 0 \)[/tex]
- Coefficient of [tex]\(x^2\)[/tex] (A): 49
- Coefficient of [tex]\(y^2\)[/tex] (C): -64
- Coefficient of the cross term [tex]\(xy\)[/tex] (B): 0
Discriminant [tex]\( \Delta = B^2 - 4AC = 0 - 4(49)(-64) = 4 \cdot 49 \cdot 64 \)[/tex]
Since [tex]\(AC < 0\)[/tex], this represents a hyperbola.
2. [tex]\( 100x^2 - 648x + 100y^2 + 200y - 6704 = 0 \)[/tex]
- Coefficient of [tex]\(x^2\)[/tex] (A): 100
- Coefficient of [tex]\(y^2\)[/tex] (C): 100
- Coefficient of the cross term [tex]\(xy\)[/tex] (B): 0
Discriminant [tex]\( \Delta = B^2 - 4AC = 0 - 4(100)(100) = -40000 \)[/tex]
Since [tex]\( \Delta < 0 \)[/tex] and [tex]\(A = C\)[/tex], this represents a circle, not a hyperbola.
3. [tex]\( 6x^2 - 12x + 54y^2 + 108y - 426 = 0 \)[/tex]
- Coefficient of [tex]\(x^2\)[/tex] (A): 6
- Coefficient of [tex]\(y^2\)[/tex] (C): 54
- Coefficient of the cross term [tex]\(xy\)[/tex] (B): 0
Discriminant [tex]\( \Delta = B^2 - 4AC = 0 - 4(6)(54) = -1296 \)[/tex]
Since [tex]\( \Delta < 0 \)[/tex], this does not represent a hyperbola.
4. [tex]\( 196x + 36y^2 + 216y - 1244 = 0 \)[/tex]
- Coefficient of [tex]\(x^2\)[/tex] (A): 0 (no [tex]\(x^2\)[/tex] term present)
- Coefficient of [tex]\(y^2\)[/tex] (C): 36
- Coefficient of the cross term [tex]\(xy\)[/tex] (B): 0
Since there is no [tex]\(x^2\)[/tex] term, this is not a quadratic equation in [tex]\(x\)[/tex] and [tex]\(y\)[/tex], hence it cannot be a hyperbola.
5. [tex]\( 4x^2 + 32x - 25y^2 - 250y + 589 = 0 \)[/tex]
- Coefficient of [tex]\(x^2\)[/tex] (A): 4
- Coefficient of [tex]\(y^2\)[/tex] (C): -25
- Coefficient of the cross term [tex]\(xy\)[/tex] (B): 0
Discriminant [tex]\( \Delta = B^2 - 4AC = 0 - 4(4)(-25) = 400 \)[/tex]
Since [tex]\(AC < 0\)[/tex], this represents a hyperbola.
6. [tex]\( 81x^2 + 512x - 64y^2 - 324y - 3836 = 0 \)[/tex]
- Coefficient of [tex]\(x^2\)[/tex] (A): 81
- Coefficient of [tex]\(y^2\)[/tex] (C): -64
- Coefficient of the cross term [tex]\(xy\)[/tex] (B): 0
Discriminant [tex]\( \Delta = B^2 - 4AC = 0 - 4(81)(-64) = 20736 \)[/tex]
Since [tex]\(AC < 0\)[/tex], this represents a hyperbola.
Therefore, the equations that represent hyperbolas are:
- [tex]\( 49x^2 - 98x - 64y^2 + 256y - 2831 = 0 \)[/tex]
- [tex]\( 4x^2 + 32x - 25y^2 - 250y + 589 = 0 \)[/tex]
- [tex]\( 81x^2 + 512x - 64y^2 - 324y - 3836 = 0 \)[/tex]
The correct equations representing hyperbolas are:
1. [tex]\( 49x^2 - 98x - 64y^2 + 256y - 2831 = 0 \)[/tex]
5. [tex]\( 4x^2 + 32x - 25y^2 - 250y + 589 = 0 \)[/tex]
6. [tex]\( 81x^2 + 512x - 64y^2 - 324y - 3836 = 0 \)[/tex]
Here, let's review the given equations one by one:
1. [tex]\( 49x^2 - 98x - 64y^2 + 256y - 2831 = 0 \)[/tex]
- Coefficient of [tex]\(x^2\)[/tex] (A): 49
- Coefficient of [tex]\(y^2\)[/tex] (C): -64
- Coefficient of the cross term [tex]\(xy\)[/tex] (B): 0
Discriminant [tex]\( \Delta = B^2 - 4AC = 0 - 4(49)(-64) = 4 \cdot 49 \cdot 64 \)[/tex]
Since [tex]\(AC < 0\)[/tex], this represents a hyperbola.
2. [tex]\( 100x^2 - 648x + 100y^2 + 200y - 6704 = 0 \)[/tex]
- Coefficient of [tex]\(x^2\)[/tex] (A): 100
- Coefficient of [tex]\(y^2\)[/tex] (C): 100
- Coefficient of the cross term [tex]\(xy\)[/tex] (B): 0
Discriminant [tex]\( \Delta = B^2 - 4AC = 0 - 4(100)(100) = -40000 \)[/tex]
Since [tex]\( \Delta < 0 \)[/tex] and [tex]\(A = C\)[/tex], this represents a circle, not a hyperbola.
3. [tex]\( 6x^2 - 12x + 54y^2 + 108y - 426 = 0 \)[/tex]
- Coefficient of [tex]\(x^2\)[/tex] (A): 6
- Coefficient of [tex]\(y^2\)[/tex] (C): 54
- Coefficient of the cross term [tex]\(xy\)[/tex] (B): 0
Discriminant [tex]\( \Delta = B^2 - 4AC = 0 - 4(6)(54) = -1296 \)[/tex]
Since [tex]\( \Delta < 0 \)[/tex], this does not represent a hyperbola.
4. [tex]\( 196x + 36y^2 + 216y - 1244 = 0 \)[/tex]
- Coefficient of [tex]\(x^2\)[/tex] (A): 0 (no [tex]\(x^2\)[/tex] term present)
- Coefficient of [tex]\(y^2\)[/tex] (C): 36
- Coefficient of the cross term [tex]\(xy\)[/tex] (B): 0
Since there is no [tex]\(x^2\)[/tex] term, this is not a quadratic equation in [tex]\(x\)[/tex] and [tex]\(y\)[/tex], hence it cannot be a hyperbola.
5. [tex]\( 4x^2 + 32x - 25y^2 - 250y + 589 = 0 \)[/tex]
- Coefficient of [tex]\(x^2\)[/tex] (A): 4
- Coefficient of [tex]\(y^2\)[/tex] (C): -25
- Coefficient of the cross term [tex]\(xy\)[/tex] (B): 0
Discriminant [tex]\( \Delta = B^2 - 4AC = 0 - 4(4)(-25) = 400 \)[/tex]
Since [tex]\(AC < 0\)[/tex], this represents a hyperbola.
6. [tex]\( 81x^2 + 512x - 64y^2 - 324y - 3836 = 0 \)[/tex]
- Coefficient of [tex]\(x^2\)[/tex] (A): 81
- Coefficient of [tex]\(y^2\)[/tex] (C): -64
- Coefficient of the cross term [tex]\(xy\)[/tex] (B): 0
Discriminant [tex]\( \Delta = B^2 - 4AC = 0 - 4(81)(-64) = 20736 \)[/tex]
Since [tex]\(AC < 0\)[/tex], this represents a hyperbola.
Therefore, the equations that represent hyperbolas are:
- [tex]\( 49x^2 - 98x - 64y^2 + 256y - 2831 = 0 \)[/tex]
- [tex]\( 4x^2 + 32x - 25y^2 - 250y + 589 = 0 \)[/tex]
- [tex]\( 81x^2 + 512x - 64y^2 - 324y - 3836 = 0 \)[/tex]
The correct equations representing hyperbolas are:
1. [tex]\( 49x^2 - 98x - 64y^2 + 256y - 2831 = 0 \)[/tex]
5. [tex]\( 4x^2 + 32x - 25y^2 - 250y + 589 = 0 \)[/tex]
6. [tex]\( 81x^2 + 512x - 64y^2 - 324y - 3836 = 0 \)[/tex]
