Question

# Find the radical axis of the pairs of circles ${x^2} + {y^2} - xy + 6x - 7y + 8 = 0$ and ${x^2} + {y^2} - xy - 4 = 0$ , the axes being inclined at ${120^ \circ }$

Hint: Make use of the equation of the radical axis of a pair of circles and solve this question
Let us consider the equation of the given two circles as ${S_1}$ and ${S_2}$.

So, we can write the equation of the first circle is given by
${S_1} = {x^2} + {y^2} - xy + 6x - 7y + 8 = 0$
The equation of the second circle is given by
${S_2} = {x^2} + {y^2} - xy - 4 = 0$
Now, we know that the equation of the radical axis of a pair of circles ${S_1}$ and ${S_2}$ is given by
${S_1} - {S_2} = 0$
So, on doing ${S_1} - {S_2}$ , we get
${x^2} + {y^2} - xy + 6x - 7y + 8 - ({x^2} + {y^2} - xy - 4) = 0$
So, on solving this we get
6x-7y+12=0
Hence the equation of the radical axis is 6x-7y+12=0.

Note: In this question we have been asked to find out the equation of radical axis of a pair of circles , in case we have been asked to find out equation of some other form of a circle then apply the suitable formula and solve it accordingly