Two circles, each of radius 5 units, touch each other at (1, 2). If the equation of their common tangents is `4x+3y=10` , find the equations of the circles.

Answer

`(x-5)^(2)+(y-5)^(2) = 5^(2)and (x+3)^(2) +(y+1)^(2) = 5^(2)`

Explanation

We have,
Slope of the common tangent `= -(4)/(3)`
`therefore` Slope of `C_(1)C_(2)=(3)/(4)`
If `C_(1)C_(2)` makes an angle `theta` with X-axis, then`costheta = (4)/(5) and sintheta = (3)/(5)`

So, the equation of `C_(1)C_(2)` in parametric form is
`(x-1)/(4//5)=(y-2)/(3//5)" "...(i)`
SInce, `C_(1) and C_(2)` are points on Eq. (i) at a distance of 5 units from P.
So, the coordinates of `C_(1) and C_(2)` are given by
`(x-1)/(4//5)=(y-2)/(3//5)=pm5rArrx=1pm4`
and `y = 2 pm 3`.
Thus, the coordinates of `C_(1) and C_(2)` are (5, 5) and (-3, -1), respectively.
Hence, the equations of the two circles are
`(x-5)^(2)+(y-5)^(2)=5^(2)`
and `(x+3)^(2)+(y+1)^(2
)=5^(2)`