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Step-by-Step Solution
Step 1: Understand the problem
We have a concave mirror with center of curvature C. An object is placed at a distance $d_1$ from C, and its image is formed at a distance $d_2$ from C. We need to find the radius of curvature $R$ of the mirror.
Step 2: Recall Newton's formula
In the context of mirrors, Newtonβs formula for a concave mirror is given by:
$$
(f + x)(f - y) = f^2
$$
where $f$ is the focal length of the mirror, $x$ is the distance of the object from the focus (on one side), and $y$ is the distance of the image from the focus (on the other side). In this problem, the object and image distances given ($d_1$ and $d_2$) are from the center of curvature C, not from the focus. However, the same relationship helps us derive the focal length when we interpret $x$ and $y$ properly.
Step 3: Express $f$ in terms of $d_1$ and $d_2$
Using the given hint and approach:
$$
(f + d_1)(f - d_2) = f^2.
$$
Expand the left-hand side:
$$
f \cdot f - f \cdot d_2 + d_1 \cdot f - d_1 \cdot d_2 = f^2.
$$
This simplifies to:
$$
f^2 + f \, d_1 - f \, d_2 - d_1 d_2 = f^2.
$$
The $f^2$ terms on both sides cancel out, leaving:
$$
f \, d_1 - f \, d_2 - d_1 d_2 = 0.
$$
Rearrange to solve for $f$:
$$
f (d_1 - d_2) = d_1 d_2
\quad\Longrightarrow\quad
f = \frac{d_1 \, d_2}{d_1 - d_2}.
$$
Step 4: Find the radius of curvature $R$
The radius of curvature $R$ of a concave mirror is related to the focal length $f$ by:
$$
R = 2f.
$$
Substitute $f = \frac{d_1 \, d_2}{d_1 - d_2}$:
$$
R = 2 \times \frac{d_1 \, d_2}{d_1 - d_2}
= \frac{2 \, d_1 \, d_2}{d_1 - d_2}.
$$
Step 5: State the final answer
Hence, the radius of curvature of the concave mirror is
$$
\boxed{\frac{2 \, d_1 \, d_2}{d_1 - d_2}}.
$$