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(D) For a convex lens to form a real image of an object on a screen separated by a fixed distance $d$, the condition for the existence of the image is

Vedclass · Accepted Answer

$\frac{d}{4}$

Vedclass · Answer

(D) For a convex lens to form a real image of an object on a screen separated by a fixed distance $d$, the condition for the existence of the image is given by the displacement method.Let $u$ be the object distance and $v$ be the image distance. Then $u + v = d$.For a real image, the lens formula is $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$.Substituting $u = -(d - v)$, we get $\frac{1}{v} + \frac{1}{d - v} = \frac{1}{f}$.This simplifies to the quadratic equation $v^2 - dv + df = 0$.For $v$ to have real roots, the discriminant must be non-negative: $D = d^2 - 4df \geq 0$.This implies $d^2 \geq 4df$, or $f \leq \frac{d}{4}$.Thus, the maximum possible focal length of the convex lens is $\frac{d}{4}$.

$A$ luminous object is separated from a screen by a distance $d$. $A$ convex lens is placed between the object and the screen such that it forms a distinct image on the screen. The maximum possible focal length of this convex lens is

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