(i) If the vertices of $\Delta A B C$ be $A(1,-3), B(4, p)$ and $C(-9,7)$ and its area is 15 square units, find the values of $p$.
(ii) The area of a triangle is $5 \mathrm{sq}$ units. Two of its vertices are $(2,1)$ and $(3,-2)$. If the third vertex is $\left(\frac{7}{2}, y\right)$, find the value of $y$.
(i) Let $A\left(x_{1}, y_{1}\right)=A(1,-3), B\left(x_{2}, y_{2}\right)=B(4, p)$ and $C\left(x_{3}, y_{3}\right)=C(-9,7)$. Now
Area $(\Delta A B C)=\frac{1}{2}\left[x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right]$
$\Rightarrow 15=\frac{1}{2}[1(p-7)+4(7+3)-9(-3-p)]$
$\Rightarrow 15=\frac{1}{2}[10 p+60]$
$\Rightarrow|10 p+60|=30$
Therefore
$\Rightarrow 10 p+60=-30$ or 30
$\Rightarrow 10 p=-90$ or $-30$
$\Rightarrow p=-9$ or $-3$
Hence, $p=-9$ or $p=-3$.
(ii)
Let $A\left(x_{1}, y_{1}\right)=A(2,1), B\left(x_{2}, y_{2}\right)=B(3,-2)$ and $C\left(x_{3}, y_{3}\right)=C\left(\frac{7}{2}, y\right)$.
Now
Area $(\Delta A B C)=\frac{1}{2}\left|x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right|$
$\Rightarrow 5=\frac{1}{2}\left|2(-2-y)+3(y-1)+\frac{7}{2}(1+2)\right|$
$\Rightarrow 10=\left|-4-2 y+3 y-3+\frac{21}{2}\right|$
$\Rightarrow 10=\left|y+\frac{7}{2}\right|$
$\Rightarrow 10=y+\frac{7}{2} \quad$ or $-10=y+\frac{7}{2}$
$\Rightarrow y=\frac{13}{2}$ or $y=\frac{-27}{2}$
Hence, $y=\frac{13}{2}$ or $\frac{-27}{2}$.