A line passing through the point (−2,0) touches the parabola y2=x−2 at the point P in the first quadrant. The area of the region bounded by the line, the parabola, and the x-axis, is:
Solutions · 1
Official solution
Write y2=4(41)(x−2); tangent line y=m(x−2)+4m1 through (−2,0):
\[0=-4m+\frac{1}{4m} \;\Rightarrow\; 16m^2=1 \;\Rightarrow\; m=\pm\frac14.\]
Taking m=41 (contact in Q1) gives contact point (6,2).
\[\text{Area}=\frac12(1)(4)+\int_2^6\left[\frac{x+2}{4}-\sqrt{x-2}\right]dx = 2+\frac23=\frac83.\]