A curve G lies in the first quadrant and passes through the point (1, 0). The tangent drawn to G at any point P meets the y-axis at the point Q. If the segment PQ has length exactly 1 for every point P on G, which of the following statements are correct? (1) y = logₑ((1 + sqrt(1 - x²))/x)

Question

A curve G lies in the first quadrant and passes through the point (1, 0). The tangent drawn to G at any point P meets the y-axis at the point Q. If the segment PQ has length exactly 1 for every point P on G, which of the following statements are correct? (1) y = logₑ((1 + sqrt(1 - x²))/x) - sqrt(1 - x²) (2) x*y' - sqrt(1 - x²) = 0 (3) y = -logₑ((1 + sqrt(1 - x²))/x) + sqrt(1 - x²) (4) x*y' + sqrt(1 - x²) = 0

Accepted Answer

Statements (1) and (4). This is the tractrix: the length of the tangent segment from the point of tangency to the y-axis is constant (= 1). The resulting differential equation is x*y' + sqrt(1 - x²) = 0, and integrating with the point (1, 0) gives y = logₑ((1 + sqrt(1 - x²))/x) - sqrt(1 - x²). Hence statements (1) and (4) are correct.

Solution

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