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# : Find k, given $A\left( 0,9 \right),B\left( 1,11 \right),C\left( 3,13 \right),D\left( 7,k \right)$ are four points and if $AB\bot CD$

Last updated date: 13th Jul 2024
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Hint: In this problem, we have to find the value of k if $AB\bot CD$. Here we can see that we are given some points, with which we can find the value of slope from the two points formula,$m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$. We will get two values of slope. We are also given that they are perpendicular, as we know that if two slope are perpendicular, then we will have the condition ${{m}_{1}}\times {{m}_{2}}=-1$, by using this condition we can find the value of k.

Complete step by step answer:
Here we have to find the value of k, if $AB\bot CD$.
We know that the given points are,
$A\left( 0,9 \right),B\left( 1,11 \right),C\left( 3,13 \right),D\left( 7,k \right)$
We can now find the value of slope from the two points formula,$m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$.
We can now find the slope value for $A\left( 0,9 \right),B\left( 1,11 \right)$
Slope of AB,
$\Rightarrow {{m}_{1}}=\dfrac{11-9}{1-0}=2$
Slope of AB, ${{m}_{1}}=2$……. (1)
We can now find the slope of $C\left( 3,13 \right),D\left( 7,k \right)$
Slope of CD,
$\Rightarrow {{m}_{2}}=\dfrac{k-13}{7-3}=\dfrac{k-13}{4}$
Slope of CD, ${{m}_{2}}=\dfrac{k-13}{4}$……… (2)
We know that if two slopes are perpendicular, then we will have the condition ${{m}_{1}}\times {{m}_{2}}=-1$.
As we have $AB\bot CD$ we can substitute (1) and (2) in the above condition, we get
$\Rightarrow 2\times \dfrac{k-13}{4}=-1$
We can now simplify the above step, we get
\begin{align} & \Rightarrow \dfrac{k-13}{2}=-1 \\ & \Rightarrow k-13=-2 \\ & \Rightarrow k=13-2=11 \\ \end{align}
Therefore, the value of k is 11.

Note: We should always remember that if two slope are perpendicular, then we will have the condition ${{m}_{1}}\times {{m}_{2}}=-1$, where we can find the value of slope as we are given four points, with the two points form $m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$. We should substitute the value of x and y correctly to get the value of the slope.