Question

If the area of the triangle with vertices $(2,5),(7,k)$ and $(3,1)$ is $10$, then the value of $k$ is

• A. $-5$ or $35$
• B. $5$ or $-35$
• C. $15$ or $-5$
• D. $-5$ or $-25$

Answer 1

The correct option is B $5$ or $-35$

Area of triangle is given by
$\frac{1}{2}\left|\begin{array}{ccc}1& 2& 5\\ 1& 7& k\\ 1& 3& 1\end{array}\right|=\pm 10$

$R_2\to R_2-R_1$ and $R_3\to R_3-R_1$
$\Rightarrow \left|\begin{array}{ccc}1& 2& 5\\ 0& 5& k-5\\ 0& 1& -4\end{array}\right|=\pm 20$

$\Rightarrow -20-k+5=\pm 20$
$\Rightarrow k=\mp 20-15$
$\Rightarrow k=-35$ or $5$