Question

If the line ax + by +c = 0 is such that a = 0 and b, c \[ \ne \] 0 then the line is perpendicular to
A. x-axis
B. y-axis
C. x + y = 1
D. x = y

Answer 1

Hint:- We had to only put the value of an equal to zero in the equation ax + by + c = 0 and then draw the graph of that equation to check the equation of line perpendicular to that.

Complete step-by-step answer:

Now the given equation is ax + by + c = 0, where a = 0 and b, c \[ \ne \] 0
Now as we know that it is given in the question that a = 0. So, the equation does not depend on the value of x because the x part of the equation will also be equal to zero.
So, the given equation becomes,
0 + by + c = 0
by + c = 0
Subtracting c from both the sides of the above equation. We get,
by = – c
On dividing both the sides of the above equation by b. We get,
\[y = \dfrac{{ - c}}{b}\]
So, the given equation becomes \[y = \dfrac{{ - c}}{b}\]. Now as we can see that y has the constant value because c and b are constant. So, now we had to check all of the options that are perpendicular to the line \[y = \dfrac{{ - c}}{b}\].
Now as we can see from the above equation that the given line has a constant value of y and the equation remains the same for each value of x because it does not depend on x.
Now as we know that x-axis is the line on the coordinate plane in which the value of y-coordinate of all the points lying on the line is equal to zero. An equation of the x-axis does not depend on the value of x. So, it will be the same for each value of x. So, the equation of x-axis will be y = 0.
y-axis is a line in the coordinate plane in which the value of x-coordinate of all the points lying on the a is equal to zero. An equation of the y-axis does not depend on the value of y. So, it will be the same for each value of y. So, the equation of y-axis will be x = 0.
Now let us plot the given equation.

So, as we can see from the above figure that the line \[y = \dfrac{{ - c}}{b}\] is perpendicular to the y-axis.
Hence, the correct option will be B.
Note:- Whenever we came up with this type of problem then we should remember that if the equation of a line is y = constant then the line must be perpendicular to y-axis and parallel to x-axis but if the equation of a line is x = constant then the line must be perpendicular to x-axis. But if the equation of line is in terms of x and y then we can check that the two lines are perpendicular by using condition that \[{m_1}{m_2} = - 1\], where \[{m_1}\] and \[{m_2}\] are the slope of two lines.