Final answer:
To find the two points on the hyperbola x² - 3y² = 9 that are closest to the point (6,0), apply the distance formula and optimization techniques using calculus, solve for minimum distances and evaluate the points that satisfy both the distance and hyperbola equations.
Step-by-step explanation:
Finding two points on the hyperbola x² - 3y² = 9 closest to the point (6,0) requires an understanding of conic sections and optimization techniques in mathematics. To solve this problem, one can apply the concept of distance formula and use the method of calculus to minimize the distance.
To begin, write the distance D from a general point on the hyperbola (x,y) to the point (6,0) using the distance formula:
D = √[(x - 6)² + (y - 0)²]
This expression becomes a function of x and y, which are not independent because they must satisfy the hyperbola equation x² - 3y² = 9. To find the minimum distance, we take the derivative of D with respect to x, and use the condition from the hyperbola to eliminate y. This leads to an optimization problem typically solved by setting the derivative equal to zero and finding critical points.
Alternatively, because we are working with a hyperbola, an analytical approach is to use the hyperbola's features, such as its vertices and foci, to find these points directly, considering symmetry and the properties of conics, but this requires deep knowledge of conic sections and might not be straightforward.
Finally, the two points closest to (6,0) will be the ones that yield the minimum distance when plugged back into the distance formula, after the optimization problem has been solved.