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3 and 4 .Determinants and Matrices
medium
જો $(2, -6), (5, 4)$ અને $(\mathrm{k}, 4)$ શિરોબિંદુવાળા ત્રિકોણનું ક્ષેત્રફળ $35$ ચોરસ એકમ હોય, તો $\mathrm{k}$ નું મૂલ્ય .............. .
A
$12$
B
$-2$
C
$12,-2$
D
$-12,-2$
Solution
The area of the triangle with vertices $(2,-6),(5,4)$ and $(k, 4)$ is given by the relation,
$\Delta=\frac{1}{2}\left|\begin{array}{ccc}2 & -6 & 1 \\ 5 & 4 & 1 \\ k & 4 & 1\end{array}\right|$
$=\frac{1}{2}[2(4-4)+6(5-k)+1(20-4 k)]$
$=\frac{1}{2}[30-6 k+20-4 k]$
$=\frac{1}{2}[50-10 k]$
$=25-5 k$
It is given that the area of the triangle id $\pm 35$.
Therefore, we have:
$\Rightarrow 25-5 k=\pm 35$
$\Rightarrow 5(5-k)=\pm 35$
$\Rightarrow 5-k=\pm 7$
When $5-k=-7, k=5+7=12$
When $5-k=-7, k=5-7=-2$
Hence, $k=12,-2$
The correct answer is $C$.
Standard 12
Mathematics
