What is the value of k if the equation of the line drawn below is ax + 7y = k?

The given equation of line is

(k−3)x−(4−k

2

)y+k

2

−7k+6=0

(a) If the given line is parallel to the x-axis, then

slope of the given line= slope of the x-axis =0

⇒

(4−k

2

)

(k−3)

=0

⇒k−3=0

⇒k=3

Thus, if the given line is parallel to the x-axis, then the value of k is 3.

(b) If the given line is parallel to the y-axis, it is vertical, hence,its slope will be undefined.

The slope of the given line is

(4−k

2

)

(k−3)

Now,

(4−k

2

)

(k−3)

is undefined at k

2

=4

⇒k=±2

Thus, if the given line is parallel to the y-axis,then the value of k is ±2.

(c), If the given line is passing through the origin, then point (0,0) satisfies the given equation of line

(k−3)(0)−(4−k

2

)(0)+k

2

−7k+6=0

⇒k

2

−7k+6=0⇒k

2

−6k−k+6=0

⇒(k−6)(k−1)=0⇒k=1 or k=6

Thus, if the given line is passing through the origin, then the value of k is either 1 or 6.

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