Question

The line $x=c$ cuts the triangle with corners $(0,0),(1,1)$ and $(9,1)$ into two regions. For the area of the two regions to be the same $c$ must be equal to

• A. $5/2$
• B. $3$
• C. $7/2$
• D. $3$ or $15$

Answer 1

The correct option is D $3$ or $15$

Equation $OB:y-1=\frac{)}{x}-5)$

$\Rightarrow$ $9y=x$

$\Rightarrow$ $E(C,\frac{C}{5})$

Area of $△OAB=\frac{1}{2}\times h\times AB=\frac{1}{2}\times 1\times 8=4$

$\Rightarrow$ Area of BDE$=\frac{4}{2}=2$

$\Rightarrow$ $\frac{1}{2}\left|\begin{array}{ccc}C& 1& 1\\ 9& 1& 1\\ C& \frac{C}{5}& 1\end{array}\right|R_1\to R_1-R_2\Rightarrow \frac{1}{2}\left|\begin{array}{ccc}C-9& 0& 1\\ 9& 1& 1\\ C& \frac{C}{5}& 1\end{array}\right|$

$=\left|\frac{1}{2}\right(C-5\left)\right(1-\frac{C}{5}|=2\Rightarrow \left|\right(C-9\left)\right(5-C\left)\right|=36\Rightarrow$

$|-{(C-9)}^2|=36\Rightarrow {(C-9)}^2=36\Rightarrow C-9=\pm 6\Rightarrow C=15or3$