Question

If (3, 2), (3, 6), (k, 9) are the collinear points . Find the value of k​

Answer 1

$k = 8 \: \ \: \: \: \: \: \: \:$

Answer 2

Step-by-step explanation:

_____________________

_______________________________

$\qquad{ \underline{ \underline{ \bf{ \red{Information \: \: from \: \: question:}}}}} \begin{cases}& \underline{ \rm{Given : }}\\ & \qquad \bullet \pmb{(3,2) \: (3,6) \: (k,9) \: \: are \: \: the \: \: collinear \: \: points \: } \\ \\ & \underline{ \rm{To \: \: Find : }} \\& \qquad \bullet \pmb{value \: \: of \: \: k \: }\end{cases}$

____________________

_____________________________

${ \underline{ \underline{ \bf{ \red{Concept :}}}}} \begin{cases} \sf \bullet{ \: If \: \: the \: \: above \: \: points \: \: are \: \: collinear \: \: they \: \: will \: \: in \: \: a \: \: same \: \: line }\\ \\ {\sf{ \bullet \: i.e, \: \: they \: \: will \: \: not \: \: form \: \: triangle }} \\ \\ \bullet\boxed{ \sf{ \green{we \: \: can \: \: say \: \: that \: \: area \: \: of \: \triangle \: \: ABC \: = 0}}} \: \: \end{cases}$

_____________________

_______________________________

${ \underline{ \underline{ \bf{ \red{Formula :}}}}} \begin{cases}{ \boxed{ \bf{ \pink{\triangle \: \: ABC = \frac{1}{2} [x _1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)]}}}} \end{cases}$

_____________________

_______________________________

${ \underline{ \underline{ \bf{ \red{Solution:}}}}}$

Let us take A(3,2) B (3,6) C (k,9)

${ \underline{ \underline{ \bf{ \red{Here : }}}}} \begin{cases} x_1 = 3 \\ y_1 = 2 \\ x_2 = 3 \\ y_2 = 6 \\ x_3 = k \\ y_3 = 9\end{cases}$

Substitute in formula ,

$\dashrightarrow \bf \: ∆ \: ABC = \frac{1}{2} (3(6 - 9) + 3(9 - 2) + k(2 - 6))$

$\dashrightarrow \bf \: ∆ \: ABC = \frac{1}{2} (3( - 3) + 3(7) + 2k - 26)$

$\dashrightarrow \bf \: ∆ \: ABC = \frac{1}{2}( - 9 + 21 - 26 + 2k)$

$\dashrightarrow \bf \: ∆ \: ABC = \frac{1}{2}( - 9 - 5 + 2k)$

$\dashrightarrow \bf \: ∆ \: ABC = \frac{1}{2}( - 14 + 2k)$

•°• Given points are collinear.

$\dashrightarrow \bf \: \frac{1}{2}( - 14 + 2k) = 0$

$\dashrightarrow \bf \:( - 14 + 2k) = 0$

$\dashrightarrow \bf \:2k = 14$

$\dashrightarrow \bf \:k = \frac{14}{2}$

$\dashrightarrow \bf \:k = 7$

•°• Value of k is 7