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Chapter 6: Problem 37

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (4,1),(4,9)\(;\) foci: (4,0),(4,10)

Short Answer

Expert verified
The standard form of the equation of the hyperbola with the given vertices and foci is \((y-5)^2/16 - (x-4)^2/9 = 1\).

Step by step solution

01

Identifying the Center

Calculate the midpoint of the line segment between the vertices to get the center. The midpoint formula is \(h = (x_1 + x_2) / 2, k = (y_1 + y_2) / 2\). Substitute the x and y values for the vertices. Therefore, \(h = (4+4) / 2 = 4, k = (1+9) / 2 = 5\). So, the center point is (4,5).
02

Calculating the Semi-Major Axis

Calculate 'a' as half of the distance between the vertices. This is simply the absolute difference of y-coordinates due to vertical orientation. Therefore, \(a = |9-1| / 2 = 4\).
03

Calculating the Semi-Minor Axis

Subsequently, calculate 'c' as half the distance between the foci. Due to vertical orientation, this is the absolute difference of y-coordinates. Therefore, \(c = |10 - 0| / 2 = 5\). Use a虏 + b虏 = c虏 to find b虏. Rearrange to get \(b^2 = c^2 - a^2\) which gives us \(b^2 = 5^2 - 4^2 = 9\).
04

Writing the Equation

Following the standard form of a vertical hyperbola equation given by \((y-h)^2/a^2 - (x-k)^2/b^2 = 1\), and substituting the parameters (h, k, a and b) derived from previous steps we get: \((y-5)^2/4^2 - (x-4)^2/9 = 1\).

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Most popular questions from this chapter

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertices: (0,卤8)\(;\) foci: (0,卤4)

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