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  4. The coordinates of the middle point of the chord cut-off by 2 x-5 y+18=0 by the circle x2+y2-6 x+2 y-54= 0 are

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Q. The coordinates of the middle point of the chord cut-off by $2 x-5 y+18=0$ by the circle $x^{2}+y^{2}-6 x+2 y-54=$ 0 are

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Solution:

The midpoint is the intersection of the chord and perpendicular line to it from the center $(3,-1)$.
The equation of perpendicular line is $5 x+2 y-13=0$.
Solving this with the given line, we get the point $(1,4)$.