We know, for a combination of a lens and a mirror,

$${P_{comb}} = 2{P_l} + {P_m}$$ ...... (1)

Also using lens maker formula,

$${1 \over {{f_l}}} = (\mu - 1)\left( {{1 \over R} - {1 \over { - R}}} \right) = (\mu - 1){2 \over R}$$

So using eq. (i),

$$ - {1 \over {{f_{comb}}}} = 2\left( {{1 \over {{f_l}}}} \right) - {1 \over {{f_m}}}$$

$$ \Rightarrow - {1 \over {{f_{comb}}}} = {{4(\mu - 1)} \over R} - {1 \over {\left( { - {R \over 2}} \right)}}$$

$$ \Rightarrow - {1 \over {{f_{comb}}}} = {{4(\mu - 1)} \over R} + {2 \over R}$$

$$ \Rightarrow - {1 \over {{f_{comb}}}} = {2 \over R}(2\mu - 2 + 1) = {2 \over R}(2\mu - 1)$$

$$ \Rightarrow {f_{comb}} = {{ - R} \over {2(2\mu - 1)}}$$

$$ \Rightarrow {R_{comb}} = 2{f_{comb}} = {{ - 2R} \over { - 2(2\mu - 1)}} = {{ - R} \over {2\mu - 1}}$$

Hence, distance $$ = {R \over {2\mu - 1}}$$