Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Connect with a community of experts ready to provide precise solutions to your questions quickly and accurately. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
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]
We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
