Question:
If \(x\) and \(y\) be the distances of the object and images formed by a concave mirror from its focus and \(f\) be the focal length then:
If \(x\) and \(y\) be the distances of the object and images formed by a concave mirror from its focus and \(f\) be the focal length then:
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Newton's Formula \(xy = f^2\) is valid for both concave and convex mirrors, provided that the distances \(x\) (object distance) and \(y\) (image distance) are strictly measured from the principal focus, not the pole.
Updated On: May 13, 2026
- \(xf = y^2\)
- \(xy = f^2\)
- \(\frac{x}{y} = f\)
- \(\frac{x}{y} = f^2\)
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The Correct Option is B
Solution and Explanation
Concept:
In ray optics, Newton's formula relates the object distance, image distance, and focal length of a spherical mirror when the distances are measured from the principal focus instead of the pole. This often provides a faster way to solve mirror problems.
Step 1: Defining the coordinates from the pole.
Let the distances measured from the pole of the concave mirror be:
• Object distance, \(u = -(f + x)\)
• Image distance, \(v = -(f + y)\)
• Focal length, \(= -f\)
Step 2: Applying the spherical mirror formula.
The standard mirror formula is: \[ \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \]
Step 3: Substituting values and simplifying.
Substituting the values from
Step 1: \[ \frac{1}{-(f+y)} + \frac{1}{-(f+x)} = \frac{1}{-f} \] Multiplying the entire equation by \(-1\): \[ \frac{1}{f+y} + \frac{1}{f+x} = \frac{1}{f} \] Taking the LCM on the left side: \[ \frac{(f+x) + (f+y)}{(f+x)(f+y)} = \frac{1}{f} \] \[ \frac{2f + x + y}{(f+x)(f+y)} = \frac{1}{f} \] Cross-multiplying: \[ f(2f + x + y) = (f+x)(f+y) \] \[ 2f^2 + fx + fy = f^2 + fy + fx + xy \] Canceling common terms (\(fx\) and \(fy\)) and subtracting \(f^2\): \[ 2f^2 - f^2 = xy \] \[ f^2 = xy \]
Step 4: Selecting the correct option.
From the calculation above, we get: \[ \boxed{xy = f^2} \] Therefore, the correct option is: \[ \boxed{\text{Option (B)}} \]
Step 1: Defining the coordinates from the pole.
Let the distances measured from the pole of the concave mirror be:
• Object distance, \(u = -(f + x)\)
• Image distance, \(v = -(f + y)\)
• Focal length, \(= -f\)
Step 2: Applying the spherical mirror formula.
The standard mirror formula is: \[ \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \]
Step 3: Substituting values and simplifying.
Substituting the values from
Step 1: \[ \frac{1}{-(f+y)} + \frac{1}{-(f+x)} = \frac{1}{-f} \] Multiplying the entire equation by \(-1\): \[ \frac{1}{f+y} + \frac{1}{f+x} = \frac{1}{f} \] Taking the LCM on the left side: \[ \frac{(f+x) + (f+y)}{(f+x)(f+y)} = \frac{1}{f} \] \[ \frac{2f + x + y}{(f+x)(f+y)} = \frac{1}{f} \] Cross-multiplying: \[ f(2f + x + y) = (f+x)(f+y) \] \[ 2f^2 + fx + fy = f^2 + fy + fx + xy \] Canceling common terms (\(fx\) and \(fy\)) and subtracting \(f^2\): \[ 2f^2 - f^2 = xy \] \[ f^2 = xy \]
Step 4: Selecting the correct option.
From the calculation above, we get: \[ \boxed{xy = f^2} \] Therefore, the correct option is: \[ \boxed{\text{Option (B)}} \]
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