Math / Algebra

QuestionWhich type of conic is represented by the equation below? $-4 x^{2}+25 y^{2}+8 x+150 y+121=0$
This is an equation of a hyperbola. Write the equation of this conic section in conic form.

Studdy Solution

STEP 1

1. The given equation is a conic section.
2. The equation is confirmed to represent a hyperbola.
3. The goal is to rewrite the equation in the standard form of a hyperbola.

STEP 2

1. Rearrange and group terms.
2. Complete the square for both $x$ and $y$ terms.
3. Rewrite the equation in standard form.

STEP 3

Rearrange and group the terms of the equation by variables:
$-4x^2 + 8x + 25y^2 + 150y + 121 = 0$
Group the $x$ terms and the $y$ terms:
$(-4x^2 + 8x) + (25y^2 + 150y) + 121 = 0$

STEP 4

Complete the square for the $x$ terms:
Factor out $-4$ from the $x$ terms:
$-4(x^2 - 2x)$
To complete the square, take half of the coefficient of $x$ , square it, and add and subtract it inside the parentheses:
$-4(x^2 - 2x + 1 - 1) = -4((x - 1)^2 - 1) = -4(x - 1)^2 + 4$

STEP 5

Complete the square for the $y$ terms:
Factor out $25$ from the $y$ terms:
$25(y^2 + 6y)$
To complete the square, take half of the coefficient of $y$ , square it, and add and subtract it inside the parentheses:
$25(y^2 + 6y + 9 - 9) = 25((y + 3)^2 - 9) = 25(y + 3)^2 - 225$

STEP 6

Substitute the completed squares back into the equation:
$-4(x - 1)^2 + 4 + 25(y + 3)^2 - 225 + 121 = 0$
Combine the constant terms:
$-4(x - 1)^2 + 25(y + 3)^2 - 100 = 0$

STEP 7

Rewrite the equation in standard form by isolating the constant:
$-4(x - 1)^2 + 25(y + 3)^2 = 100$
Divide the entire equation by 100 to get the standard form of a hyperbola:
$\frac{(y + 3)^2}{4} - \frac{(x - 1)^2}{25} = 1$
This is the equation of the hyperbola in standard form.

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QuestionWhich type of conic is represented by the equation below? $-4 x^{2}+25 y^{2}+8 x+150 y+121=0$
This is an equation of a hyperbola. Write the equation of this conic section in conic form.

Studdy Solution

STEP 1

1. The given equation is a conic section.
2. The equation is confirmed to represent a hyperbola.
3. The goal is to rewrite the equation in the standard form of a hyperbola.

STEP 2

1. Rearrange and group terms.
2. Complete the square for both $x$ and $y$ terms.
3. Rewrite the equation in standard form.

STEP 3

Rearrange and group the terms of the equation by variables:
$-4x^2 + 8x + 25y^2 + 150y + 121 = 0$
Group the $x$ terms and the $y$ terms:
$(-4x^2 + 8x) + (25y^2 + 150y) + 121 = 0$

STEP 4

Complete the square for the $x$ terms:
Factor out $-4$ from the $x$ terms:
$-4(x^2 - 2x)$
To complete the square, take half of the coefficient of $x$ , square it, and add and subtract it inside the parentheses:
$-4(x^2 - 2x + 1 - 1) = -4((x - 1)^2 - 1) = -4(x - 1)^2 + 4$

STEP 5

Complete the square for the $y$ terms:
Factor out $25$ from the $y$ terms:
$25(y^2 + 6y)$
To complete the square, take half of the coefficient of $y$ , square it, and add and subtract it inside the parentheses:
$25(y^2 + 6y + 9 - 9) = 25((y + 3)^2 - 9) = 25(y + 3)^2 - 225$

STEP 6

Substitute the completed squares back into the equation:
$-4(x - 1)^2 + 4 + 25(y + 3)^2 - 225 + 121 = 0$
Combine the constant terms:
$-4(x - 1)^2 + 25(y + 3)^2 - 100 = 0$

STEP 7

Rewrite the equation in standard form by isolating the constant:
$-4(x - 1)^2 + 25(y + 3)^2 = 100$
Divide the entire equation by 100 to get the standard form of a hyperbola:
$\frac{(y + 3)^2}{4} - \frac{(x - 1)^2}{25} = 1$
This is the equation of the hyperbola in standard form.

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QuestionWhich type of conic is represented by the equation below? −4x2+25y2+8x+150y+121=0-4 x^{2}+25 y^{2}+8 x+150 y+121=0−4x2+25y2+8x+150y+121=0This is an equation of a hyperbola. Write the equation of this conic section in conic form.

Studdy Solution

STEP 1

1. The given equation is a conic section.2. The equation is confirmed to represent a hyperbola.3. The goal is to rewrite the equation in the standard form of a hyperbola.

STEP 2

1. Rearrange and group terms.2. Complete the square for both xxx and yyy terms.3. Rewrite the equation in standard form.

STEP 3

Rearrange and group the terms of the equation by variables:−4x2+8x+25y2+150y+121=0 -4x^2 + 8x + 25y^2 + 150y + 121 = 0 −4x2+8x+25y2+150y+121=0Group the xxx terms and the yyy terms:(−4x2+8x)+(25y2+150y)+121=0 (-4x^2 + 8x) + (25y^2 + 150y) + 121 = 0 (−4x2+8x)+(25y2+150y)+121=0

STEP 4

STEP 5

STEP 6

Substitute the completed squares back into the equation:−4(x−1)2+4+25(y+3)2−225+121=0 -4(x - 1)^2 + 4 + 25(y + 3)^2 - 225 + 121 = 0 −4(x−1)2+4+25(y+3)2−225+121=0Combine the constant terms:−4(x−1)2+25(y+3)2−100=0 -4(x - 1)^2 + 25(y + 3)^2 - 100 = 0 −4(x−1)2+25(y+3)2−100=0

STEP 7

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QuestionWhich type of conic is represented by the equation below? $-4 x^{2}+25 y^{2}+8 x+150 y+121=0$
This is an equation of a hyperbola. Write the equation of this conic section in conic form.

1. The given equation is a conic section.
2. The equation is confirmed to represent a hyperbola.
3. The goal is to rewrite the equation in the standard form of a hyperbola.

1. Rearrange and group terms.
2. Complete the square for both $x$ and $y$ terms.
3. Rewrite the equation in standard form.

Rearrange and group the terms of the equation by variables:
$-4x^2 + 8x + 25y^2 + 150y + 121 = 0$
Group the $x$ terms and the $y$ terms:
$(-4x^2 + 8x) + (25y^2 + 150y) + 121 = 0$

Substitute the completed squares back into the equation:
$-4(x - 1)^2 + 4 + 25(y + 3)^2 - 225 + 121 = 0$
Combine the constant terms:
$-4(x - 1)^2 + 25(y + 3)^2 - 100 = 0$