A ray of light generated from the source kept at (-3,4) and strikes the line `2x + y = 7` at R. and then teminated at (0,1) . Find the point R so that ray travels through the shortest distance.

Answer

To find the point R where the ray of light strikes the line \(2x + y = 7\) such that the distance traveled is minimized, we can follow these steps:

### Step 1: Identify the Points
The source of light is at point \(A(-3, 4)\) and the ray terminates at point \(B(0, 1)\). The line equation is given as \(2x + y = 7\).

### Step 2: Parameterize the Point R
Let the coordinates of point R be \(R(x, y)\). Since R lies on the line \(2x + y = 7\), we can express \(y\) in terms of \(x\):
\[ y = 7 - 2x \]

### Step 3: Write the Distance Function
We need to find the distance \(AR + RB\). The distance \(AR\) from point A to point R is given by:
\[ AR = \sqrt{(x + 3)^2 + (y - 4)^2} \]
Substituting \(y = 7 - 2x\):
\[ AR = \sqrt{(x + 3)^2 + ((7 - 2x) - 4)^2} = \sqrt{(x + 3)^2 + (3 - 2x)^2} \]

The distance \(RB\) from point R to point B is:
\[ RB = \sqrt{(x - 0)^2 + (y - 1)^2} = \sqrt{x^2 + (y - 1)^2} \]
Substituting \(y = 7 - 2x\):
\[ RB = \sqrt{x^2 + ((7 - 2x) - 1)^2} = \sqrt{x^2 + (6 - 2x)^2} \]

### Step 4: Total Distance Function
The total distance \(D\) is:
\[ D = AR + RB = \sqrt{(x + 3)^2 + (3 - 2x)^2} + \sqrt{x^2 + (6 - 2x)^2} \]

### Step 5: Minimize the Distance
To minimize \(D\), we can differentiate \(D\) with respect to \(x\) and set the derivative equal to zero. This process involves using calculus to find critical points.

1. Differentiate \(D\) with respect to \(x\).
2. Set the derivative equal to zero and solve for \(x\).

### Step 6: Solve for y
Once we find the value of \(x\), substitute it back into \(y = 7 - 2x\) to find the corresponding \(y\) coordinate.

### Step 7: Conclusion
The coordinates of point R will be \((x, y)\) where \(x\) and \(y\) are the values obtained from the previous steps.