Consider a curved mirror y = f(x) passing through (8, 6) having the property that all light rays emerging from origin, after reflected from the mirror becomes parallel to x-axis. The equation of the mirror is `y^(a) = b(c-x^(d))` where a, b, c, d are constants, then

Options

A. b=4
B. b = 36
C. c= 9
D. c = 1

Answer

To solve the problem, we need to find the constants \( a, b, c, d \) in the equation of the mirror given by \( y^a = b(c - x^d) \) such that the mirror reflects light rays from the origin (0,0) parallel to the x-axis and passes through the point (8,6).

### Step 1: Understanding the Reflection Condition
The problem states that all light rays emerging from the origin become parallel to the x-axis after reflecting off the mirror. This implies that the mirror is a parabola opening to the right, which can be represented in the form \( y^2 = 4px \) for some constant \( p \).

### Step 2: Relating the Given Equation to the Parabola
The given equation of the mirror is \( y^a = b(c - x^d) \). To match the form of a parabola, we can assume \( a = 2 \) and \( d = 1 \), which gives us:
\[ y^2 = b(c - x) \]
This resembles the standard form of a parabola.

### Step 3: Finding Constants \( b \) and \( c \)
Now, we need to find \( b \) and \( c \) such that the mirror passes through the point (8, 6). Substituting \( x = 8 \) and \( y = 6 \) into the equation:
\[ 6^2 = b(c - 8) \]
This simplifies to:
\[ 36 = b(c - 8) \quad \text{(1)} \]

### Step 4: Analyzing the Reflection Condition
For the light rays to reflect parallel to the x-axis, we can derive that the vertex of the parabola should be at the origin, which leads us to the conclusion that \( b \) must be related to the focal length of the parabola.

### Step 5: Finding Values for \( b \) and \( c \)
We can also express \( b \) in terms of \( c \) and substitute it back into equation (1). However, we need more information to find specific values. The problem suggests testing various options for \( b \) and \( c \).

### Step 6: Testing Options
Given the options provided in the video transcript, we can test the combinations for \( b \) and \( c \):
1. Let’s test \( b = 36 \) and \( c = 9 \):
\[ 36 = 36(9 - 8) \implies 36 = 36 \quad \text{(True)} \]
This combination satisfies the equation.

### Conclusion
Thus, the constants are:
- \( a = 2 \)
- \( b = 36 \)
- \( c = 9 \)
- \( d = 1 \)

### Final Answer
The values of the constants are:
- \( a = 2 \)
- \( b = 36 \)
- \( c = 9 \)
- \( d = 1 \)