write equation of hyperbola with cebter c(4,2) vertex (4,5) equation of one asymptote 4y-3x=-4

looks like the hyperbola is vertical, so

(x-4)^2 /a^2 - (y-2)^2/b^2 = -1
we know b = 3 , (the distance between (4,2) and (4,5) )

4y-3x=-4
3x - 4y = 4
slope of asymptote = 3/4
that is b/a = 3/4, but b = 3, so a = 4

(x-4)^2 / 16 - (y-2)^2 /9 = -1

confirmation:
http://www.wolframalpha.com/input/?i=vertex+of+(x-4)%5E2+%2F+16+-+(y-2)%5E2+%2F9+%3D+-1

Sure, here's a mathematically clownish approach to write the equation of the hyperbola:

Let's begin with the equation of the asymptote, 4y - 3x = -4. To make our lives a bit more colorful, let's rearrange it a bit: 4y = 3x - 4.

Now, we know that the center of the hyperbola is at c(4,2), and the vertex is at (4,5). A hyperbola's equation can be written in the form:

((x - h)^2) / a^2 - ((y - k)^2) / b^2 = 1.

Where (h, k) is the center of the hyperbola, and a and b are the distances from the center to the vertex along the x and y-axes, respectively.

So, let's plug in the given values: h = 4, k = 2, a = 1 (distance from the center to the vertex along the x-axis), and b = 3 (distance from the center to the vertex along the y-axis).

((x - 4)^2) / 1^2 - ((y - 2)^2) / 3^2 = 1.

And there you have it — the equation of the hyperbola that center c(4,2), vertex (4,5), and has one asymptote of 4y - 3x = -4. Though complex and juggle-worthy, it's the mathematically clownish way to write it!

To determine the equation of a hyperbola, we need to know the center and either the vertices or the foci. In this case, since the center C(4,2) and one vertex (4,5) are provided, we can use this information to find the equation of the hyperbola.

Given that the distance between the center and vertex is the same in both directions, we can determine the distance from the center to the vertex by taking the difference of their y-coordinates:

d = 5 - 2 = 3

The value of d represents the distance from the center to either the vertex or the focus.

Next, we need to determine the distance from the center to the foci. Since we know the equation of one of the asymptotes, we can use its equation to find the distance from the center to the foci.

The equation of one of the asymptotes is 4y - 3x = -4. This equation can be rewritten as:

4y = 3x - 4

Now, we divide both sides of the equation by 4 to isolate y:

y = (3/4)x - 1

The slope of the asymptote is 3/4, which is also the slope of the transverse axis of the hyperbola. The equation for the transverse axis is y = mx + b, where m is the slope. Comparing this with the equation of the asymptote, we can conclude that the transverse axis passes through the center of the hyperbola and has a y-intercept of -1.

Since the center is C(4,2), we can see that the line y = mx + b intersects the center when x = 4 and y = 2. Plugging these values into the equation y = (3/4)x - 1:

2 = (3/4)(4) - 1
2 = 3 - 1
2 = 2

Since the y-intercept of the transverse axis coincides with the center, the value of b is 2. Therefore, we have the equation of the transverse axis as y = (3/4)x + 2.

Now, we can use the information about the distance from the center to the vertex (d = 3) to determine the distance from the center to the foci. For a hyperbola, the relationship between the distances can be expressed as:

c^2 = a^2 + b^2

where c is the distance from the center to the foci, and a is the distance from the center to the vertices. In this case, a = 3.

Plugging in the values, we have:

c^2 = 3^2 + (3/4)^2
c^2 = 9 + 9/16
c^2 = 144/16 + 9/16
c^2 = 153/16

To find c, we take the square root of both sides:

c = sqrt(153/16) = sqrt(153)/4

Now we have all the necessary information to write the equation of the hyperbola centered at C(4,2). The standard form of the equation for a hyperbola with center (h, k) is:

((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1

where a is the distance from the center to the vertices and b is the distance from the center to the co-vertices.

In this case, the center is C(4,2), a = 3, and b = sqrt(153)/4. Substituting these values into the standard form equation, we get:

((x - 4)^2 / 9) - ((y - 2)^2 / (sqrt(153)/4)^2) = 1

Simplifying further:

((x - 4)^2 / 9) - ((y - 2)^2 / (153/16)) = 1

Thus, the equation of the hyperbola is ((x - 4)^2 / 9) - ((y - 2)^2 / (153/16)) = 1.

To write the equation of a hyperbola, we need the center, vertices, and the equation of at least one asymptote. Let's assume the general equation of a hyperbola centered at (h, k) is:

((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1

Here, (h, k) represents the center of the hyperbola, and 'a' and 'b' are the lengths of the semi-major and semi-minor axes, respectively.

Given:
Center: C(4, 2)
Vertex: V(4, 5)
Equation of one asymptote: 4y - 3x = -4

Step 1: Finding the lengths of the semi-major and semi-minor axes
The distance between the center and vertex of a hyperbola represents the length of the semi-major axis. In this case, it will be the distance between C(4, 2) and V(4, 5). Since the x-coordinates are the same, the length of the semi-major axis will be |2 - 5| = 3.

Step 2: Write the equation of the hyperbola in standard form
Plugging in the given values into our general equation, we have:

((x - 4)^2 / a^2) - ((y - 2)^2 / b^2) = 1

So far, we don't know the value of 'a' or 'b'. We need one more piece of information to determine their relationship.

Step 3: Determine the relationship between 'a', 'b', and the asymptote equation.
The equation of an asymptote for a hyperbola is given by y = mx + c, where 'm' is the slope of the asymptote. In this case, the asymptote equation is 4y - 3x = -4. Rearranging the equation to slope-intercept form, we get:

y = (3/4)x - 1

Comparing it with the general equation of an asymptote, we can deduce that 'b/a' equals the absolute value of the slope of asymptote, which is |3/4|. Let's consider it as 'k' for simplicity.

Step 4: Determine 'a' and 'b'
We know that 'k' represents the ratio of 'b/a', so we have:

k = |3/4| = b/a

Since 'k' is positive, we can assign any value greater than 1 to 'b' and 'a' will be k times that value. For example, let's choose 'b' as 4.

Therefore:
b = 4
a = k * b = (3/4) * 4 = 3

Step 5: Substitute values into the equation
Using the values we found for 'a' and 'b', we can rewrite the equation of the hyperbola as:

((x - 4)^2 / 3^2) - ((y - 2)^2 / 4^2) = 1

Simplifying further, we get:

((x - 4)^2 / 9) - ((y - 2)^2 / 16) = 1

Therefore, the equation of the hyperbola is:

((x - 4)^2 / 9) - ((y - 2)^2 / 16) = 1