Question

The tangent and the normal drawn to the curve $y=x^2-x+4$ at $P(1,4)$ cut the $x$-axis at $A$ and $B$, respectively. Then, the area of the $\Delta PAB$ in sq unit is

• A. $4$
• B. $32$
• C. $8$
• D. $16$

Answer 1

Answer: (D)

$16$

Given, equation of curve is $y=x^2-x+4$
Slope of tangent $P(1,4)$ is
$\left(\frac{dy}{dx}\right)=2x-1$
$\Rightarrow {\left(\frac{dy}{dx}\right)}_{(1,4)}=2-1=1$
Equation of tangent is
$y-4=1(x-1)$
$\Rightarrow y-x=3$ ....(i)
and equation of normal at point $P(1,4)$ is
$y-4=-1(x-1)$
$\Rightarrow x+y=5$ ....(ii)
Since, the tangent cuts $x$-axis at $A$
Therefore, coordinates of $A$ are $(-3,0)$ and the normal cuts $x$-axis at $B$ and coordinates of $B$ are $(5,0)$.
Therefore, area of $\Delta PAB$
$=\frac{1}{2}\left|\begin{array}{ccc}1& 4& 1\\ -3& 0& 1\\ 5& 0& 1\end{array}\right|$
$=\frac{1}{2}\left|[-4(-3-5)]\right|$
$=\frac{1}{2}\left|32\right|$
$=16$ sq units