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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 \[AB\bot CD\]

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Last updated date: 13th Jul 2024
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Answer
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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.