Final answer:
To find the equation of an ellipse with foci at (±3,0) that passes through the point (4,1), we solve for the semi-major axis (a) using the distance formula and then find the semi-minor axis (b) using the relationship a²=b²+c². The equation of the ellipse is then derived in the standard form.
Step-by-step explanation:
The student is asking to find the equation of an ellipse with given foci at (±3,0) that passes through the point (4,1). To solve this, we need to use the definition of an ellipse, which is the set of all points for which the sum of the distances to the two foci is constant.
Step-by-step Solution:
- Let the constant sum of the distances to the foci be 2a, where a is the semi-major axis of the ellipse.
- The distance between the foci is 2c, where c is the distance from the center of the ellipse to a focus. Since the foci are at (±3,0), this means c = 3.
- The point (4,1) must satisfy the condition that the sum of its distances to the foci is 2a. Using the distance formula, we can set up the equation: √((4-(-3))² + (1-0)²) + √((4-3)² + (1-0)²) = 2a.
- Solving the above equation for a gives us the length of the semi-major axis.
- Using the relationship between a, b (the semi-minor axis), and c, which is a² = b² + c², we can solve for b.
- Finally, the standard form of the equation of an ellipse centered at the origin with a horizontal major axis is (x^2/a^2) + (y^2/b^2) = 1.
Inserting our calculated values of a and b into this formula gives us the equation of the ellipse.