Final answer:
To find the equation of the hyperbola with given foci and a point, we can use the standard form and the distance formula to determine the values of a, b, and find the equation.
Step-by-step explanation:
To find the equation of the hyperbola, we can use the given information about its foci and a point it passes through. The foci are at (0, ±√14) and the point P(3,4) is on the hyperbola. We can start by finding the equation in the standard form (x - h)²/a² - (y - k)²/b² = 1 for a hyperbola with its center at (h,k). In this case, the center is at (0,0).
Next, we can use the distance formula to determine the values of a and b: |PF₁ - PF₂| = 2a, where PF₁ and PF₂ are the distances from the foci to a point on the hyperbola. Solving this equation will give us the values of a and b. Finally, we can substitute the values of a, b, h, and k into the standard form equation to find the equation of the hyperbola.
Therefore, the equation of the hyperbola with foci at (0, ±√14) and passing through the point P(3,4) is x²/14 - y²/10 = 1.