Equation of line joining mathrm{A}(1,3) and mathrm{B}(0,0) using determinants and find mathrm{k} if

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3 and 4 .Determinants and Matrices

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Find the equation of the line joining $\mathrm{A}(1,3)$ and $\mathrm{B}(0,0)$ using determinants and find $\mathrm{k}$ if $\mathrm{D}(\mathrm{k}, 0)$ is a point such that area of triangle $\mathrm{ABD}$ is $3 \,\mathrm{sq}$ $\mathrm{units}$.

A

$\mp 5$

B

$\mp 2$

C

$\mp 7$

D

$\mp 9$

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., $k=\mp 2$

Standard 12

Mathematics

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