Question

Variation of $v$ with $u$ for a spherical mirror is as shown in figure. This curve is a hyperbola. A straight line of unit slope intersects the hyperbola at point 'A'. If the focal length of mirror is 20 cm, coordinates of point A are:

• A. 20 cm, 10 cm
• B. 40 cm, 40 cm
• C. 40 cm, 20 cm
• D. 20 cm, 20 cm

Answer 1

Answer: (B)

40 cm, 40 cm

Answer 2

The mirror formula is given by $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$. However, the problem states that the variation of $v$ with $u$ is a hyperbola, and the figure shows this hyperbola in the first quadrant with asymptotes along the $u$ and $v$ axes. This suggests the equation of the hyperbola is of the form $\frac{1}{u} + \frac{1}{v} = \frac{1}{f}$. Given the focal length $f = 20$ cm, the equation becomes $\frac{1}{u} + \frac{1}{v} = \frac{1}{20}$.

A straight line of unit slope intersects the hyperbola at point 'A'. The figure indicates this line passes through the origin and makes an angle of $45^\circ$ with the $u$-axis, meaning its slope is $\tan(45^\circ) = 1$. Thus, the equation of the line is $v = u$.

To find the coordinates of point 'A', we find the intersection of $\frac{1}{u} + \frac{1}{v} = \frac{1}{20}$ and $v = u$. Substituting $v=u$ into the hyperbola equation: $\frac{1}{u} + \frac{1}{u} = \frac{1}{20}$ $\frac{2}{u} = \frac{1}{20}$ $u = 2 \times 20 = 40$ cm.

Since $v=u$, then $v = 40$ cm. Therefore, the coordinates of point 'A' are $(40 \text{ cm}, 40 \text{ cm})$.