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Question: : Find k, given \[A\left( 0,9 \right),B\left( 1,11 \right),C\left( 3,13 \right),D\left( 7,k \right)\...

: Find k, given A(0,9),B(1,11),C(3,13),D(7,k)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⊥CDAB\bot CD

Explanation

Solution

In this problem, we have to find the value of k if AB⊥CDAB\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=y2−y1x2−x1m=\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 m1×m2=−1{{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⊥CDAB\bot CD.
We know that the given points are,
A(0,9),B(1,11),C(3,13),D(7,k)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=y2−y1x2−x1m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}.
We can now find the slope value for A(0,9),B(1,11)A\left( 0,9 \right),B\left( 1,11 \right)
Slope of AB,
⇒m1=11−91−0=2\Rightarrow {{m}_{1}}=\dfrac{11-9}{1-0}=2
Slope of AB, m1=2{{m}_{1}}=2……. (1)
We can now find the slope of C(3,13),D(7,k)C\left( 3,13 \right),D\left( 7,k \right)
Slope of CD,
⇒m2=k−137−3=k−134\Rightarrow {{m}_{2}}=\dfrac{k-13}{7-3}=\dfrac{k-13}{4}
Slope of CD, m2=k−134{{m}_{2}}=\dfrac{k-13}{4}……… (2)
We know that if two slopes are perpendicular, then we will have the condition m1×m2=−1{{m}_{1}}\times {{m}_{2}}=-1.
As we have AB⊥CDAB\bot CD we can substitute (1) and (2) in the above condition, we get
⇒2×k−134=−1\Rightarrow 2\times \dfrac{k-13}{4}=-1
We can now simplify the above step, we get

& \Rightarrow \dfrac{k-13}{2}=-1 \\\ & \Rightarrow k-13=-2 \\\ & \Rightarrow k=13-2=11 \\\ \end{aligned}$$ Therefore, the value of k is 11. **Note:** We should always remember that if two slope are perpendicular, then we will have the condition $${{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.