Step 1: Understanding the Concept:
The study of spherical mirrors usually relies on the standard mirror formula, which measures distances from the pole \( (P) \) of the mirror.
However, Sir Isaac Newton formulated an alternative approach that measures distances of the object and image from the principal focus \( (F) \) of the mirror.
This is known as Newton's Formula for spherical mirrors and lenses.
In this framework, let \( x \) be the distance of the object from the focus and \( y \) be the distance of the image from the focus.
The focal length \( f \) is the distance between the pole and the focus.
This method often simplifies algebra, especially in problems where the focus is used as a reference point.
For a concave mirror, the focus is real and lies in front of the reflecting surface.
Step 2: Key Formula or Approach:
The standard mirror formula used in ray optics is:
\[ \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \]
Where:
\( u = \) Object distance from the pole.
\( v = \) Image distance from the pole.
\( f = \) Focal length of the mirror.
According to the Newtons convention:
\( u = f + x \) (Magnitude-wise)
\( v = f + y \) (Magnitude-wise)
Step 3: Detailed Explanation:
Let us apply the Cartesian sign convention for a concave mirror.
The pole is the origin. The focus is at \( (-f, 0) \).
The object is placed at a distance \( x \) from the focus, so its position is \( u = -(f + x) \).
The image is formed at a distance \( y \) from the focus, so its position is \( v = -(f + y) \).
The focal length of a concave mirror is negative, so \( f_{mirror} = -f \).
Now, substitute these into the mirror equation:
\[ \frac{1}{-(f + y)} + \frac{1}{-(f + x)} = \frac{1}{-f} \]
Multiply the whole equation by \( -1 \):
\[ \frac{1}{f + y} + \frac{1}{f + x} = \frac{1}{f} \]
To combine the fractions on the left side, find the common denominator:
\[ \frac{(f + x) + (f + y)}{(f + y)(f + x)} = \frac{1}{f} \]
Simplify the numerator and expand the denominator:
\[ \frac{2f + x + y}{f^2 + fx + fy + xy} = \frac{1}{f} \]
Cross-multiply the terms across the equals sign:
\[ f(2f + x + y) = f^2 + fx + fy + xy \]
\[ 2f^2 + fx + fy = f^2 + fx + fy + xy \]
Notice that \( fx \) and \( fy \) appear on both sides of the equation.
Subtracting \( fx \) and \( fy \) from both sides:
\[ 2f^2 = f^2 + xy \]
Subtract \( f^2 \) from both sides to isolate the product \( xy \):
\[ f^2 = xy \]
This derived equation \( xy = f^2 \) is the standard form of Newton's formula for mirrors.
It shows that the product of the distances measured from the focus is always equal to the square of the focal length.
Step 4: Final Answer:
The mathematical relationship between the focal length and the distances from the focus is \( xy = f^2 \).
This matches Option (B).