Problem 18 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Foci: (±2,0)\(;\) major axis of length 10

Step-by-Step Solution

Answer

The equation of the ellipse is \(\frac{x^{2}}{25} + \frac{y^{2}}{21} = 1\).

Step 1: Finding the value of c and a

The foci of an ellipse are \((\pm c, 0)\), so given that the foci are \((\pm2,0)\), we know that \(c = 2\). Also, the length of the major axis is 10, which is twice the distance from the center to the ellipse, or \(2a\). Therefore, \(a = \frac{10}{2} = 5\).

Step 2: Solve for \(b^{2}\)

The relationship between \(a, b, c\) for an ellipse is given by \(c^{2} = a^{2} - b^{2}\). Substituting our known values we get \(2^{2} = 5^{2} - b^{2}\). Solving this equation for \(b^{2}\), we find that \(b^{2} = 5^{2} - 2^{2} = 21\).

Step 3: Formulating the equation

Now that we have the values for \(a^{2}\) and \(b^{2}\), we can substitute these into the general form of the ellipse equation, which is \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\). Hence the equation of the ellipse is \(\frac{x^{2}}{5^{2}} + \frac{y^{2}}{21} = 1\).