Question

The circle $x^2+y^2-6x-10y+k=0.$ does not touch or intersect the co-ordinate axes and the point (1,4) is inside the circle. find the ranges of the values of k.

Answer 1

S = x${}^2$ + y${}^2$ - 6x - 10y + k = 0.
Centre C(3, 5), r${}^2$ = 9 + 25 k = 34 - k.
Point P (1, 4) lies inside the circle therefore
// S= -ive // - 29 + k < 0 or k < 29 .(1)
Since the circle neither touches nor intersects the axes of co-ordinates, therefore from the figure it is clear that r < 3 (xco-ordinate of centre) and r < S(y co-ordinate of centre).
Hence, r < 3 $\Rightarrow$ r${}^2$ < 9
or 34 - k < 9 or 25 < k .(2)
From (1) and (2), we have 25 < k < 29.