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Let c be a constant with c > 1. A line through the point (1, c) with slope m bounds, together with the parabola y = x², a finite region. If the least possible area of this region (minimised over m) equals 36 square units, find the value of c² + m² for that minimising line.

1. 104
2. 81
3. 100
4. 121

Correct answer: 104

Solution

For the line y = m(x-1)+c meeting y = x², the area enclosed is (1/6)*(difference of roots)³. The squared difference of roots is m² - 4m + 4c, minimised at m = 2 giving 4c - 4. Setting the area equal to 36 yields (4c-4)^(3/2) = 216, so 4c - 4 = 36 and c = 10. Hence c² + m² = 100 + 4 = 104. (Note: the original option set did not contain the correct value, so plausible options including 104 have been supplied.)

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