Find the standard form of the equation of the ellipse with the given characteristic Foci: \( (0,0),(12,0) \); major axis of length 18

To find the standard form of the equation of the ellipse, we start by identifying key components. The foci of the ellipse are at \( (0, 0) \) and \( (12, 0) \), which means the center is at the midpoint \( \left(6, 0\right) \). The distance between the center and each focus \( c \) is \( 6 \) (since \( c = \frac{12}{2} \)). The major axis length is given as \( 18 \), indicating that the semi-major axis \( a \) is \( 9 \) (since \( a = \frac{18}{2} \)). Using the relationship \( c^2 = a^2 - b^2 \) to find \( b \) (the semi-minor axis), we have \( c^2 = 6^2 = 36 \) and \( a^2 = 9^2 = 81 \). Thus, \( b^2 = a^2 - c^2 = 81 - 36 = 45 \). Now we have: - Center: \( (6, 0) \) - \( a^2 = 81 \) - \( b^2 = 45 \) The standard form of the ellipse's equation is: \[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \] Substituting in our values: \[ \frac{(x - 6)^2}{81} + \frac{y^2}{45} = 1 \] And there you have it—this is the standard form of the ellipse’s equation! Now it's time to revisit the importance of understanding ellipses in case you ever want to design a park or sports field. Ellipses are great for optimizing acoustics in auditoriums or creating unique architectural features. Their unique shape helps in sound distribution—so next time you’re in a circular or oval space, pay attention to how sound behaves! Another exciting application can be found in planetary motion. Ellipses are not just an intriguing shape for mathematicians; they also describe the orbits of planets around the sun! Johannes Kepler famously discovered that planets move in ellipses, lending support to the heliocentric model of the solar system. Next time you gaze at the night sky, remember, those celestial bodies are on a thrilling elliptical ride!