Question

The area of the region bounded by the parabola $(y-2)^2=x-1$ , the tangent to the parabola at the point $(2, 3)$ and the $x$-axis is

• A. $3$
• B. $6$
• C. $9$
• D. $12$

Answer 1

Answer: (C)

$9$

Answer 2

Equation of tangent at (2, 3) to $(y-2)^2=x-1$ is $S_1=0$ $? x - 2y + 4 = 0$ Required Area = Area of $?OCB +$ Area of $OAPD -$ Area of $?PCD$ $=\frac{1}{2}\left(4\times2\right)+$ $\int\limits^{3}_{{0}}(y^2-4y+5)dy-$ $\frac{1}{2}\left(1\times2\right)$ $=4+\left[\frac{y^{3}}{3}-2y^{2}+5y\right]^{^{^3}}_{_{_0}}-1=4-9-18+15+-1$ $=28-19=9$ s units (or) Area $=\int\limits^{3}_{{0}}(2y-4-y^2+4y-5)dy-\int\limits^{3}_{{0}}(y^2+6y-5)dy=-$ $\int\limits^{3}_{{0}}(3-y)^2\,dy$ $=\left[\frac{\left(y-3\right)^{3}}{3}\right]^{^{^3}}_{_{_0}}=\frac{27}{3}=9$ sunits