MathDB

In the coordinate plane with

O(0,\ 0)

, consider the function C: \ y \equal{} \frac 12x \plus{} \sqrt {\frac 14x^2 \plus{} 2} and two distinct points

P_1(x_1,\ y_1),\ P_2(x_2,\ y_2)

C

. (1) Let H_i\ (i \equal{} 1,\ 2) be the intersection points of the line passing through P_i\ (i \equal{} 1,\ 2), parallel to

x

axis and the line y \equal{} x. Show that the area of

\triangle{OP_1H_1}

and

\triangle{OP_2H_2}

are equal. (2) Let

x_1 < x_2

. Express the area of the figure bounded by the part of

x_1\leq x\leq x_2

for

C

and line segments

P_1O,\ P_2O

in terms of

y_1,\ y_2

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