Question

Area of triangle whose vertices are $\left(a,a^2\right),\left(b,b^2\right)$ and ($c,c^2$ ) is $\frac{1}{2},$ and area of another triangle whose vertices are $\left(p,p^2\right),\left(q,q^2\right)$ and $\left(r,r^2\right)$ is $4$, then the value of
$\left|\begin{array}{ccc}{\left(1+ap\right)}^2& {\left(1+bp\right)}^2& {\left(1+cp\right)}^2\\ {\left(1+aq\right)}^2& {\left(1+bq\right)}^2& {\left(1+cq\right)}^2\\ {\left(1+ar\right)}^2& {\left(1+br\right)}^2& {\left(1+cr\right)}^2\end{array}\right|$ is

• A. 2
• B. 4
• C. 8
• D. 16

Answer 1

Answer: (D)

$\left|\begin{array}{ccc}{\left(1+ap\right)}^2& {\left(1+bp\right)}^2& {\left(1+cp\right)}^2\\ {\left(1+aq\right)}^2& {\left(1+bq\right)}^2& {\left(1+cq\right)}^2\\ {\left(1+ar\right)}^2& {\left(1+br\right)}^2& {\left(1+cr\right)}^2\end{array}\right|$
$=\left|\begin{array}{ccc}1& 2a& a^2\\ 1& 2b& b^2\\ 1& 2c& c^2\end{array}\right|\times \left|\begin{array}{ccc}1& p& p^2\\ 1& q& q^2\\ 1& r& r^2\end{array}\right|$
$=2\times 2{\Delta }_1\cdot 2{\Delta }_2$
$=8{\Delta }_1{\Delta }_2=8\times \frac{1}{2}\times 4=16$