given the eccentilisty and a point that lies on the hyperbola ,find the equation of the hyperbola center origin
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To find the equation of a hyperbola with the center at the origin, given the eccentricity and a point that lies on the hyperbola, follow these steps:
Step 1: Identify the given information:
- Eccentricity (e)
- Coordinates of a point on the hyperbola (x1, y1)
Step 2: Determine the coordinates of the focus (F) using the eccentricity:
- The distance from the origin (O) to the focus (F) is given by the formula: |OF| = ae, where "a" is the distance between the center and a vertex.
- Since the center is at the origin (0, 0), the distance |OF| = ae simplifies to |F| = ae.
- Therefore, the coordinates of the focus are (ae, 0).
Step 3: Determine the coordinates of the given point (x1, y1).
Step 4: Calculate the distance from the point to the focus:
- The distance from the given point (x1, y1) to the focus (ae, 0) is given by the formula: √[(x1 - ae)^2 + (y1 - 0)^2].
Step 5: Use the distance formula for a hyperbola to create the equation:
[(x - h)^2 / a^2] - [(y - k)^2 / b^2] = 1, where (h, k) is the center of the hyperbola.
- Since the center is at the origin, the equation becomes:
x^2 / a^2 - y^2 / b^2 = 1
Step 6: Substitute the known values into the equation:
- Substitute the coordinates of the given point (x1, y1) and the coordinates of the focus (ae, 0) into the distance formula from Step 4.
- The equation should now have only three unknowns: a, b, and e.
Step 7: Solve for a, b, and e by simplifying the equation:
- Simplify the equation by expanding and rearranging it in terms of a, b, and e.
- You can then solve for a, b, and e using algebraic methods or by applying other known facts about the hyperbola (e.g., the relationship between a, b, and e).
By following these steps, you can find the equation of a hyperbola with the center at the origin, given the eccentricity and a point that lies on the hyperbola.