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First focal length and second focal length of a lens will be the same only when

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radii of curvature of both surfaces are some

The refractive indices of the media on both sides of the lens are the same.

The lens is plano convex or plano concave

The lens is equiconvex or equiconcave and refractive indices of the media on both sides of the lens are the same.

answer is D.

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Detailed Solution

General lens equation is

$\frac{μ_{3}}{V} - \frac{μ_{1}}{u} = (\frac{μ_{2} - μ_{1}}{R_{1}} - \frac{μ_{2} - μ_{3}}{R_{2}})$

For first focal length $V = a and u = - f_{1}$

$∴ \frac{1}{f_{1}} = \frac{(μ_{2} - μ_{1}) / μ_{1}}{R_{1}} - \frac{(μ_{2} - μ_{3}) μ_{1}}{R_{2}}$

For second focal length, $u = α and V = + f_{2}$

$∴ \frac{1}{f_{2}} = \frac{(μ_{2} - μ_{1}) / μ_{3}}{R_{1}} - \frac{(μ_{2} - μ_{3}) / μ_{3}}{R_{2}}$

Obviously when $μ_{1} = μ_{3}$

$f_{1} will be equal to f_{2}$

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