okay so the question is as follows:

determien the standard form of the equation of a hyperbola with vertices(+or- 2,0) and passing through (4,3)

is this right?
x^2/2^2 - y^2/b^2 =1
Since P(4,3) passes through, the x and y coordinates satisfy the equation:

4^2/2^2 - 3^2/b^2 =1

b^2=3

thus the equation is
x^2/4 -y^2/3=1

is this right?

I see no errors.

thank you!

Explain Bot Explain Bot answered
8 months ago

Yes, your solution is correct. You have correctly determined the standard form of the equation of a hyperbola with vertices at (+/-2,0) and passing through (4,3).

To find the standard form of the equation of a hyperbola, you need to use the information given about the vertices and a point on the hyperbola. The standard form of a hyperbola equation is:

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

where (h,k) represents the center of the hyperbola, and a and b represent the distances from the center to the vertices along the x-axis and y-axis, respectively.

In your case, the vertices are (+/-2,0), so the center of the hyperbola is at (0,0) since the vertices are located on the x-axis.

The distance from the center to the vertices along the x-axis is 2, so a=2.

To find the value of b, we can use the point on the hyperbola (4,3). Plugging these values into the equation, we get:

(4-0)^2/2^2 - (3-0)^2/b^2 = 1
16/4 - 9/b^2 = 1

Simplifying this equation, we get:

4 - 9/b^2 = 1

From this equation, we can solve for b^2:

- 9/b^2 = 1 - 4
- 9/b^2 = -3
b^2 = 3

Now that we have the values of a^2 and b^2, we can substitute them back into the standard form equation:

(x-0)^2/2^2 - (y-0)^2/√3^2 = 1
x^2/4 - y^2/3 = 1

Therefore, the standard form of the equation of the hyperbola is x^2/4 - y^2/3 = 1.

Good job on solving the question correctly!

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