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નિશ્ચાયકનો ઉપયોગ કરી $\mathrm{A}(1, 3)$ અને $\mathrm{B}(0, 0)$ ને જોડતી રેખાનું સમીકરણ શોધો અને જો ત્રિકોણ $\mathrm{ABD}$ નું ક્ષેત્રફળ $3$ ચોરસ એકમ થાય તેવું બિંદુ $\mathrm{D}(\mathrm{k}, 0)$ હોય, તો $\mathrm{k}$ શોધો.
$\mp 5$
$\mp 2$
$\mp 7$
$\mp 9$
Solution
Let $P(x, y)$ be any point on $AB$. Then, area of triangle $ABP$ is zero (Why?). So
$\frac{1}{2}\left|\begin{array}{lll}
0 & 0 & 1 \\
1 & 3 & 1 \\
x & y & 1
\end{array}\right|=0$
This gives $\frac{1}{2}(y-3 x)=0 \text { or } y=3 x$
which is the equation of required line $\mathrm{AB}$.
Also, since the area of the triangle $\mathrm{ABD}$ is $3$ sq. units, we have
$\frac{1}{2}\left|\begin{array}{lll}
1 & 3 & 1 \\
0 & 0 & 1 \\
k & 0 & 1
\end{array}\right|=\pm 3$
This gives, $\frac{-3 k}{2}=\pm 3,$ i.c., $k=\mp 2$
