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In the standard equation for a hyperbola that opens up and down the value b equals half the length of the hyperbola's transverse axis?
true
Which expression gives the length go the transverse axis of the hyperbola shown below?
The length of the transverse axis of a hyperbola is determined by the value of (2a), where (a) is the distance from the center to each vertex along the transverse axis. In the standard forms of hyperbolas, such as ((x-h)^2/a^2 - (y-k)^2/b^2 = 1) or ((y-k)^2/a^2 - (x-h)^2/b^2 = 1), (a) represents this distance. Therefore, to find the length of the transverse axis, you would use the expression (2a).
Where is the transverse axis for the hyperbola y2 -25x2 100?
To find the transverse axis of the hyperbola given by the equation ( y^2 - 25x^2 = 100 ), we first rewrite it in standard form: ( \frac{y^2}{100} - \frac{x^2}{4} = 1 ). This equation indicates that the hyperbola is oriented vertically, with its center at the origin (0, 0). The transverse axis is vertical and extends along the y-axis, with its length determined by the value of ( a ) (which is 10 in this case, since ( a^2 = 100 )). Thus, the transverse axis is along the line ( y = \pm 10 ).
What expression gives the length of the transverse axis of the hyperbola?
The length of the transverse axis of a hyperbola is given by the expression ( 2a ), where ( a ) is the distance from the center of the hyperbola to each vertex along the transverse axis. For a hyperbola centered at the origin with the standard form ( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 ) (horizontal transverse axis) or ( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 ) (vertical transverse axis), the value of ( a ) determines the extent of the transverse axis. Thus, the transverse axis length varies directly with ( a ).
What is the equation for a hyperbola with transverse axis of length 24 and centered at the origin?
The standard form of the equation of a hyperbola with center at the origin isx2/a2 - y2/b2 = 1 where the transverse axis lies on the x-axis,ory2/a2 - x2/b2 = 1 where the transverse axis lies on the y-axis.The vertices are a units from the center and the foci are c units from the center.For both equations, b2 = c2 - a2. Equivalently, c2 = a2 + b2.Since we know the length of the transverse axis (the distance between the vertices), we can find the value of a (because the center, the origin, lies midway between the vertices and foci).Suppose that the transverse axis of our hyperbola lies on the x-axis.Then, |a| = 24/2 = 12So the equation becomes x2/144 - y2/b2 = 1.To find b we need to know what c is.
In the standard equation for a hyperbola that opens up and down the value b equals half the length of the hyperbola's transverse axis?
true
Which expression gives the length go the transverse axis of the hyperbola shown below?
The length of the transverse axis of a hyperbola is determined by the value of (2a), where (a) is the distance from the center to each vertex along the transverse axis. In the standard forms of hyperbolas, such as ((x-h)^2/a^2 - (y-k)^2/b^2 = 1) or ((y-k)^2/a^2 - (x-h)^2/b^2 = 1), (a) represents this distance. Therefore, to find the length of the transverse axis, you would use the expression (2a).
Where is the transverse axis for the hyperbola y2 -25x2 100?
To find the transverse axis of the hyperbola given by the equation ( y^2 - 25x^2 = 100 ), we first rewrite it in standard form: ( \frac{y^2}{100} - \frac{x^2}{4} = 1 ). This equation indicates that the hyperbola is oriented vertically, with its center at the origin (0, 0). The transverse axis is vertical and extends along the y-axis, with its length determined by the value of ( a ) (which is 10 in this case, since ( a^2 = 100 )). Thus, the transverse axis is along the line ( y = \pm 10 ).
What is the definition and equation of rectangular hyperbola?
Defn: A hyperbola is said to be a rectangular hyperbola if its asymptotes are at right angles. Std Eqn: The standard rectangular hyperbola xy = c2
What expression gives the length of the transverse axis of the hyperbola?
The length of the transverse axis of a hyperbola is given by the expression ( 2a ), where ( a ) is the distance from the center of the hyperbola to each vertex along the transverse axis. For a hyperbola centered at the origin with the standard form ( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 ) (horizontal transverse axis) or ( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 ) (vertical transverse axis), the value of ( a ) determines the extent of the transverse axis. Thus, the transverse axis length varies directly with ( a ).
What is the equation for a hyperbola with transverse axis of length 24 and centered at the origin?
The standard form of the equation of a hyperbola with center at the origin isx2/a2 - y2/b2 = 1 where the transverse axis lies on the x-axis,ory2/a2 - x2/b2 = 1 where the transverse axis lies on the y-axis.The vertices are a units from the center and the foci are c units from the center.For both equations, b2 = c2 - a2. Equivalently, c2 = a2 + b2.Since we know the length of the transverse axis (the distance between the vertices), we can find the value of a (because the center, the origin, lies midway between the vertices and foci).Suppose that the transverse axis of our hyperbola lies on the x-axis.Then, |a| = 24/2 = 12So the equation becomes x2/144 - y2/b2 = 1.To find b we need to know what c is.
How is a hyperbola made?
A hyperbola is formed by the intersection of a double cone with a plane that cuts through both halves of the cone, but is not parallel to the cone's axis. This results in two separate curves, known as branches, that open away from each other. The mathematical definition of a hyperbola involves the difference in distances from any point on the curve to two fixed points, called foci, being constant. Hyperbolas can also be described using their standard equation in Cartesian coordinates.
What are the Properties of rectangular hyperbola?
A rectangular hyperbola is a specific type of hyperbola where the transverse and conjugate axes are equal in length, making it symmetrical about both axes. Its standard equation is (xy = c^2), where (c) is a constant. Key properties include that its asymptotes are perpendicular to each other, and it has a unique feature of having equal distances from the center to the vertices along the axes. Additionally, the slopes of the tangent lines at any point on the hyperbola are negative reciprocals of each other, reflecting its symmetry.
What does each variable represent in standard form?
There are different standard forms for different things. There is a standard form for scientific notation. There is a standard form for the equation of a line, circle, ellipse, hyperbola and so on.
How do you write the generic equation suggested by a graph showing a hyperbola?
To write the generic equation of a hyperbola suggested by a graph, first identify the orientation of the hyperbola (horizontal or vertical) based on its shape. The standard forms are ((x - h)^2/a^2 - (y - k)^2/b^2 = 1) for a horizontal hyperbola and ((y - k)^2/a^2 - (x - h)^2/b^2 = 1) for a vertical hyperbola, where ((h, k)) is the center, and (a) and (b) are distances that determine the shape. Use points on the graph to find the values of (h), (k), (a), and (b) to complete the equation.
What is the major difference in the equation of a hyperbola compared to the equation of an ellipse?
The major difference between the equations of a hyperbola and an ellipse lies in the signs of the terms. In the standard form of an ellipse, both squared terms have the same sign (positive), resulting in a bounded shape. In contrast, the standard form of a hyperbola has a difference in signs (one positive and one negative), which results in two separate, unbounded branches. This fundamental difference in sign leads to distinct geometric properties and behaviors of the two conic sections.
How do you write the equation for each hyperbola in standard form when the equation is y2-3x2 plus 6x plus 6y equals 18?
y2-3x2+6x+6y= 18 is in standard form. The vertex form would be (y+3)2/24 - (x-1)2/8 = 1
