Find the image without losing a sign.

A small illuminated arrow stands 30 cm to the left of an ideal converging thin lens with focal length 12 cm. Light travels left to right. Use the Cartesian convention: distances to the right of the lens are positive.

Where is the image, and what is its signed lateral magnification?

Organize the givens & assumptions

Object coordinate u: −30 cm
Focal length f: +12 cm

Paraxial rays, thin lens in air, real object on the incoming-light side. Measure all distances from the lens; use upward-positive heights for both object and image.

A $v=+20\ \mathrm{cm},\ m=-\frac23$
B $v=-20\ \mathrm{cm},\ m=+\frac23$
C $v=+20\ \mathrm{cm},\ m=+\frac23$
D $v=+\frac{60}{7}\ \mathrm{cm},\ m=+\frac27$

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