How to carry out overall replacement?

Question

There is an equation:

(-3 + 4 k^2)/(-6 + m^2) == 1

and a fraction:

Sqrt[m^2]/Sqrt[1 + k^2]

How can the equation be used to replace m ^ 2 in the fraction?

The expected result is:

Sqrt[4 k^2 + 3]/Sqrt[1 + k^2]

edit1: thank you Syed！If the formula contains x0 and x0 ^ 2, how to replace only the x0 ^ 2 items in the formula, and keep x0 without replacing.

eqn = x0^2/a^2 + y0^2/b^2 == 1

a fraction:

y - y0 == (x - x0) (-x0 y0 + Sqrt[-a^2 b^2 + b^2 x0^2 + a^2 y0^2])/( a^2 - x0^2)

use:

y - y0 == (x - x0) (-x0 y0 + Sqrt[-a^2 b^2 + b^2 x0^2 + a^2 y0^2])/(
    a^2 - x0^2) /. Solve[eqn,x0^2] // FullSimplify

Tt is running error

My problem is to use equation eqn to replace x0 ^ 2 in the formula, and x0 is not replaced. How to operate, the desired results are:

y0 (-x x0 + a^2 (1 - (y y0)/b^2))

Answer 1

One possible way:

eq1 = (-3 + 4 k^2)/(-6 + m^2) == 1;
Sqrt[m^2]/Sqrt[1 + k^2] /. Solve[eq1, m]

$$\left\{\frac{\sqrt{4 k^2+3}}{\sqrt{k^2+1}},\frac{\sqrt{4 k^2+3}}{\sqrt{k^2+1}}\right\}$$