A patient's kidney stone is placed 24 units away from the source of the shock waves of a lithotripter. The lithotripter is based on an ellipse with a minor axis that measures 18 units. Find an equation of an ellipse that would satisfy this situation. An equation of this ellipse with center $(0,0)$ is
Solution
Step 1 ：A patient's kidney stone is placed 24 units away from the source of the shock waves of a lithotripter. The lithotripter is based on an ellipse with a minor axis that measures 18 units. We are asked to find an equation of an ellipse that would satisfy this situation.
Step 2 ：An equation of this ellipse with center at the origin (0,0) is given by the formula: \((x^2/a^2) + (y^2/b^2) = 1\), where \(a\) is the semi-major axis and \(b\) is the semi-minor axis.
Step 3 ：In this case, the semi-major axis \(a\) is the distance from the source of the shock waves to the kidney stone, which is 24 units. The semi-minor axis \(b\) is half the length of the minor axis, which is 9 units.
Step 4 ：Substituting the values of \(a\) and \(b\) into the equation, we get: \((x^2/24^2) + (y^2/9^2) = 1\).
Step 5 ：Simplifying this, we get the equation of the ellipse: \((x^2/576) + (y^2/81) = 1\).
Step 6 ：Final Answer: The equation of the ellipse is \(\boxed{(x^2/576) + (y^2/81) = 1}\).
