Final answer:
The product of the perpendicular distances from the foci to any tangent to the ellipse x²/a² + y²/b² = 1 is a² - b². This is due to a property of ellipses regarding the distances from the foci to a tangent line.
Step-by-step explanation:
The product of the perpendicular distances from the foci to any tangent to the ellipse x²/a² + y²/b² = 1 is a² - b². To understand why, we first recognize that an ellipse is a closed curve such that the sum of the distances from a point on the curve to the two foci is constant. This constant sum is equal to the length of the major axis, which is 2a.
Considering the general property that for any tangent to an ellipse, the product of the distances from the foci to the tangent is equal to the square of the distance from the center to the tangent along the major axis. This distance is given by the semi-major axis, a.
The foci of an ellipse are c distance away from the center, where c² = a² - b², and since they are symmetrically located, the distances from the foci to any tangent line multiply to c². Thus, replacing c with its value, we get the product of the distances as a² - b², which corresponds to option B.