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.
To find the value of k, we need to make use of the given information that the circle passes through the point of intersection of the coordinate axes and the two lines.
Let's start by finding the point of intersection of the coordinate axes. This point will have x-coordinate 0 and y-coordinate 0 since it lies on both the x-axis and the y-axis. So, the coordinates of this point are (0, 0).
Next, let's find the point of intersection of the lines kx - y + 1 = 0 and x - 2y + 3 = 0. To do this, we can solve the two equations simultaneously.
Solve the first equation for y:
kx - y + 1 = 0
y = kx + 1
Substitute this value of y into the second equation:
x - 2(kx + 1) + 3 = 0
x - 2kx - 2 + 3 = 0
x - 2kx + 1 = 0
Combine like terms:
(1 - 2k)x + 1 = 0
For this equation to hold true, the coefficient of x should be zero since the left side does not depend on x. So, we have:
1 - 2k = 0
Solve this equation for k:
2k = 1
k = 1/2
Therefore, the value of k is 1/2.