Question

Area of a triangle with vertices $(a,b)$, $(x_1,y_1)$ and $(x_2,y_2)$ where a, $x_1$, $x_2$ with in G.P. with common ratio r and b, $y_1$, $y_2$ are in G.P. with common ratio s, is

• A. $ab(r-1)(s-1)(s-r)$
• B. $\frac{1}{2}ab(r+1)(s+1)(s-r)$
• C. $\frac{1}{2}ab(r-1)(s-1)(s-r)$
• D. $ab(r+1)(s+1)(r-s)$

Answer 1

The correct option is C $\frac{1}{2}ab(r-1)(s-1)(s-r)$

From given we have,

$x_1=ar,x_2=a{r}^2$

$y_1=bs,y_2=b{s}^2$

The area of triangle is given by,

$\Delta =\frac{1}{2}\left|\begin{array}{ccc}a& b& 1\\ x_1& y_1& 1\\ x_2& y_2& 1\end{array}\right|$

$\Delta =\frac{1}{2}\left|\begin{array}{ccc}a& b& 1\\ ar& bs& 1\\ a{r}^2& b{s}^2& 1\end{array}\right|$

$\Delta =\frac{1}{2}ab\left|\begin{array}{ccc}1& 1& 1\\ r& s& 1\\ r^2& s^2& 1\end{array}\right|$

$C_1\to C_1-C_3,C_2\to C_2-C_3$

$\Delta =\frac{1}{2}ab\left|\begin{array}{ccc}0& 0& 1\\ r-1& s-1& 1\\ r^2-1& s^2-1& 1\end{array}\right|$

$\Delta =\frac{1}{2}ab(r-1)(s-1)\left|\begin{array}{ccc}0& 0& 1\\ 1& 1& 1\\ r^2-1& s^2-1& 1\end{array}\right|$

$\therefore \Delta =\frac{1}{2}ab(r-1)(s-1)(s-r)$