The graph that shows the correct variation of $\frac{1}{v}$ with $\frac{1}{u}$ for a concave mirror, where $u$

Question

The graph that shows the correct variation of $\frac{1}{v}$ with $\frac{1}{u}$ for a concave mirror, where $u$ is the object distance and $v$ is the image distance, is:

Accepted Answer

1.  **Mirror Formula:** The relationship between object distance ($u$), image distance ($v$), and focal length ($f$) for a spherical mirror is given by the mirror formula: $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$.
2.  **Equation of the Line:** Rearranging this formula to plot $\frac{1}{v}$ on the y-axis and $\frac{1}{u}$ on the x-axis gives $\frac{1}{v} = -\frac{1}{u} + \frac{1}{f}$. This represents the equation of a straight line, $y = mx + c$, where the slope $m = -1$ and the y-intercept $c = \frac{1}{f}$.
3.  **Sign Convention:** For a concave mirror, the focal length $f$ is negative. Therefore, both the y-intercept ($y = \frac{1}{f}$) and the x-intercept ($x = \frac{1}{f}$ when $y=0$) must be on the negative axes.
4.  **Conclusion:** The correct graph is a straight line with a slope of $-1$ that passes through the negative x-axis and negative y-axis (i.e., a straight line in the second, third, and fourth quadrants intersecting the negative axes).

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