For a concave mirror prove that r = 2f

1.	For a concave mirror prove that r = 2f
Answer» Answer : Assuming that a ray of LIGHT is coming from infinite distance and strikes on the concave mirror directly. (Ray of light is parallel to the MAIN axis) We know that, Any incident ray TRAVELING parallel to the principal axis will pass through the focal point after reflection. Now see the attachment carefully! :) ❒ Question is totally based on geometry only. For ∆CMN : $\sf:\implies\:tan\:i=\dfrac{MN}{CN}$ $\sf:\implies\:MN=tan\:i\cdot CN$ Here ANGLE i is small, Therefore, tan i ≈ i $\sf:\implies\:MN=i\cdot CN\:\dots\:(I)$ For ∆FMN : $\sf:\implies\:tan\:2i=\dfrac{MN}{FN}$ $\sf:\implies\:MN=tan\:2i\cdot FN$ Applying same condition, $\sf:\implies\:MN=2i\cdot FN\:\dots\:(II)$ By equating both equations, we get $\sf:\implies\:i\cdot CN=2i\cdot FN$ $\sf:\implies\:CN=2FN$ CN = R = Radius of curvature FN = f = Focal length $:\implies\:\underline{\boxed{\bf{\gray{R=2f}}}}$

Answer»