Question

A piece of cheese is located at \[\left( 12,10 \right)\] in the coordinate plane. A mouse is at \[\left( 4,-2 \right)\] and is running up the line \[y=-5x+18\]. At the point \[\left( a,b \right)\] the mouse starts getting farther from the cheese rather than closer to it. The value of \[\left( a+b \right)\] is
A. 6
B. 10
C. 18
D. 14

Answer 1

Hint: In this problem, we have to find the value of \[\left( a+b \right)\], where At the point \[\left( a,b \right)\] the mouse starts getting farther from the cheese rather than closer to it. We are given that a piece of cheese is located at \[\left( 12,10 \right)\] in the coordinate plane and the mouse is at \[\left( 4,-2 \right)\] and is running up the line \[y=-5x+18\]. We can first draw, mark the points and the line in the coordinate plane. We will have the slope value for the given line equation, we can then find the slope for its perpendicular line and substitute it in the slope formula to find the value of a and b and we can find \[\left( a+b \right)\].

Complete step by step solution:
Here a piece of cheese is located at \[\left( 12,10 \right)\] in the coordinate plane. A mouse is at \[\left( 4,-2 \right)\] and is running up the line \[y=-5x+18\]. At the point \[\left( a,b \right)\] the mouse starts getting farther from the cheese rather than closer to it. We have to find the value of \[\left( a+b \right)\].
We can first mark the given points and the line equation, we get

We know that the given equation is \[y=-5x+18\]……. (1)
Where the equation is of the form \[y=mx+c\] where m is the slope,
Hence the slope of BC, m = -5
We know that, when the slope is given then the slope of its perpendicular is \[{{m}_{1}}=-\dfrac{1}{m}\].
Hence, the slope of AC, \[{{m}_{1}}=\dfrac{1}{5}\].
We know that, we have the point \[\left( 12,10 \right)\] and the slope \[{{m}_{1}}=\dfrac{1}{5}\].
We can now use the slope formula with points, to find the value of a and b.
\[\Rightarrow {{m}_{1}}=\dfrac{y-{{y}_{1}}}{x-{{x}_{1}}}\]
We can now substitute the above values, we get
\[\begin{align}
  & \Rightarrow \dfrac{1}{5}=\dfrac{y-10}{x-12} \\
 & \Rightarrow x-12=5y-50 \\
 & \Rightarrow x=5y-38......(2) \\
\end{align}\]
 We can now substitute (1) in (2), we get
\[\begin{align}
  & \Rightarrow x=5\left( -5x+18 \right)-38 \\
 & \Rightarrow x=-25x+90-38 \\
 & \Rightarrow 26x=52 \\
 & \Rightarrow x=2 \\
\end{align}\]
We can now substitute x = 2 in (2), we get
\[\Rightarrow y=-10+18=8\]
The value of \[\left( a,b \right)=\left( 2,8 \right)\], then
\[\Rightarrow a+b=2+8=10\]
The value of \[\left( a+b \right)\] is 10.
Therefore, the answer is option B. 10.

Note: We should always remember that the formula of slope intercept form is \[y=mx+c\], where m is the slope and c is the y-intercept. We should also know that when the slope is given then the slope of its perpendicular is \[{{m}_{1}}=-\dfrac{1}{m}\]. We should remember that the formula for slope when a point is given is \[{{m}_{1}}=\dfrac{y-{{y}_{1}}}{x-{{x}_{1}}}\].