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
$\begin{vmatrix}\left(1+ap\right)^{2}&\left(1+bp\right)^{2}&\left(1+cp\right)^{2}\\ \left(1+aq\right)^{2}&\left(1+bq\right)^{2}&\left(1+cq\right)^{2}\\ \left(1+ar\right)^{2}&\left(1+br\right)^{2}&\left(1+cr\right)^{2}\end{vmatrix}$ is

• A. 2
• B. 4
• C. 8
• D. 16

Answer 1

Answer: (D)

$\begin{vmatrix}\left(1+ap\right)^{2}&\left(1+bp\right)^{2}&\left(1+cp\right)^{2}\\ \left(1+aq\right)^{2}&\left(1+bq\right)^{2}&\left(1+cq\right)^{2}\\ \left(1+ar\right)^{2}&\left(1+br\right)^{2}&\left(1+cr\right)^{2}\end{vmatrix}$
$=\begin{vmatrix}1&2a&a^{2}\\ 1&2b&b^{2}\\ 1&2c&c^{2}\end{vmatrix}\times \begin{vmatrix}1&p&p^{2}\\ 1&q&q^{2}\\ 1&r&r^{2}\end{vmatrix}$
$=2\times2\Delta_{1}\cdot2\Delta_{2}$
$=8\Delta_{1}\Delta_{2}=8\times\frac{1}{2}\times4=16$