if a circle passes through the point of intersection of the co-ordinate axes with the lines kx-y+1=0 and x-2y+3=0. find k.

Notice that kx - y + 1 = 0 and x - 2y + 3 = 0 have different y intercepts because

y = kx -1 for the first line and

y = 1/2x - 3/2 for the second line

Therefore they can only intersect the coordinate axes on the X axis. That is where the x - intercept is.

The x - intercept is found where y = 0

Therefore, for the first second line:

0 = 1/2x -3/2

1/2x = 3/2

x = 3

The first line MUST have the same x - intercept, so we plug in x = 3 and solve for k:

0 = kx - 1

0 = k(3) -1

3k = 1

k = 1/3

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