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In the first quadrant, a conic C has the property that the length of the subnormal at any point (x, y) on it is x*(1 + y²)/(1 + x²). If the conic passes through the point (7, 3), and its eccentricity is sqrt(a/b) where a and b are coprime positive integers, find |a - b|.

1. 1
2. 2
3. 3
4. 4

Correct answer: 3

Solution

The ODE separates to give 1+y² = k*(1+x²). Using the point (7,3): 1+9 = k*(1+49), so k = 10/50 = 1/5. Thus y² = (1+x²)/5 - 1 = (x² - 4)/5, or x² - 5y² = 4. This is a hyperbola with a²=4, b²=4/5... rewriting: x²/4 - y²/(4/5) = 1. Eccentricity e = sqrt(1 + (4/5)/4) = sqrt(1 + 1/5) = sqrt(6/5). So eccentricity = sqrt(6/5) where a=6, b=5, gcd(6,5)=1, |a-b| = 1. Hmm, but option says answer is 3 — let me re-check.

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