question_answer

The area of the region bounded by the parabola \[{{(y-2)}^{2}}=x-1\], the tangent of the parabola at the point (2, 3) and the x-axis is:

A) 6

B) 9

C) 12

D) 3

Correct Answer: B

Solution :

[b] The given parabola is \[{{(y-2)}^{2}}=x-1\]

Vertex (1, 2) and it meets x - axis at (5, 0)

Also it gives \[{{y}^{2}}-4y-x+5=0\]

So, that equation of tangent to the parabola at (2, 3) is

\[y.3-2(y+3)-\frac{1}{2}(x+2)+5=0\] or \[x-2y+4=0\]

which meets x-axis at (-4, 0).

In the figure shaded area is the required area.

Let us draw PD perpendicular to y - axis.

Then required area \[=Ar\,\,\Delta BOA+Ar(OCPD)\]

\[-Ar(\Delta APD)\]

\[=\frac{1}{2}\times 4\times 2+\int_{0}^{3}{xdy-\frac{1}{2}\times 2\times 1=}\]

\[3+\int_{0}^{3}{{{(y-2)}^{2}}+1dy}\]

\[=3+\left[ \frac{{{(y-2)}^{3}}}{3}+y \right]_{0}^{3}=3+\left[ \frac{1}{3}+3+\frac{8}{3} \right]\]

\[=3+6=9\] Sq. units