Find the values of k for which the line `(k-3)x-(4-k^2)` `y+k^2-7k+6=0` is (a) Parallel to the xaxis, (b) Parallel to the vaxis, (c) Passing through the origin.

Explanation

Let m be the slope of the line
`(k-3(x-(4-k^(2))+k^(2)-7k+6=0" "...(i)`
Then, `m=(-(k-3))/(-(-4-k^(2)))=(k-3)/(4-k^(2))`
(a) If the line is parallel to x-axis, then
Slope `=0implies(k-3)/(4-k^(2))=0impliesk-3=0impliesk=3`
(b) If the line is parallel to y-axis, then
`1/m=0implies(4-k^(2))/(4-3)=0 implies4-k^(2)=0impliesk=+-2`
(c ) If the line passes through the origin, then (0,0) must satisfy the equation (i)
`implies(k-3)xx0-(4-k^(2))xx0+k^(2)-7k+6=0`
`implies(k-1)(k-6)=0impliesk=1,6`