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A straight line of negative slope passes through the point (3, 4) and…
Question
A straight line of negative slope passes through the point (3, 4) and meets the positive x-axis at A and the positive y-axis at B. Find the minimum possible area of triangle OAB, where O is the origin.
✓ Verified answer: 24checked by our engine — not a guess
Step-by-step solution
Write the line in intercept form x/a + y/b = 1 with a, b > 0.
Passing through (3,4) gives 3/a + 4/b = 1, so b = 4a/(a-3) (requires a > 3).
The triangle area is A = ab/2 = 2a^2/(a-3).
Differentiating: dA/da = 2(2a(a-3) - a^2)/(a-3)^2 = 2(a^2 - 6a)/(a-3)^2 = 0 gives a = 6.
Then b = 24/3 = 8 and A = (6)(8)/2 = 24.
The second derivative is positive, so this is a minimum.
Final answer24
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