Question

If the graph of the function \[y=mx+c\] passes through the origin, then ‘c’ must be equal to zero.
(A) False
(B)Sometimes true
(C) Always true
(D) Intermediate

Answer 1

Hint: The equation of the given function is the same as the equation of a straight line. Since the straight line is passing through the origin, it must satisfy the coordinate of the origin. The coordinate of the origin is \[(0,0)\] . Put this coordinate in the equation \[y=mx+c\] and then, solve it further.

Complete step by step solution:
According to the question, it is given that the function \[(y=mx+c)\] passes through the origin.
We also know that a straight line has the equation of the form \[y=mx+c\] where m is the slope of the straight line and c is the y-intercept of the straight line.
So, the equation of the function \[y=mx+c\] is the same as the equation straight line. Therefore, we can say that the given function is a straight line.
\[y=mx+c\] ………………….(1)
Since it is given that the function passes through the origin so, it must satisfy the coordinate of the origin.
The coordinate of the origin is \[(0,0)\] .
Putting x=0 and y=0 in equation (1), we get
\[\begin{align}
  & y=mx+c \\
 & \Rightarrow 0=m.0+c \\
 & \Rightarrow 0=0+c \\
 & \Rightarrow 0=c \\
\end{align}\]
So, the value of c is zero and also in the equation of a straight line \[y=mx+c\] , we have c as the intercept of the straight line.
According to the question, our statement is if the graph of the function \[y=mx+c\] passes through the origin, then ‘c’ must be equal to zero. And also, after solving we get c which is equal to zero.
Therefore, the given statement is true.
Hence the correct option is option (C).

Note: We can also solve this question in another way. We know the equation of a straight line passing through the origin.

The equation passing through the origin is \[y=mx\] …………………(1)
According to the question we have the equation of the straight line which is \[y=mx+c\] and this straight line is also passing through the origin.
But we know that the equation of the straight line \[y=mx+c\] has y intercept equal to c. That is, the straight line is intersecting the y axis at point A and the coordinate of point A is \[\left( 0,c \right)\] . Since the line \[y=mx+c\] is passing through the origin so, the point A must coincide with the origin.

\[y=mx+c\] ………………….(2)
Equation (1) and Equation (2) must be equal to each other.
On comparing, we get,
\[\begin{align}
  & mx=mx+c \\
 & \Rightarrow 0=c \\
\end{align}\]
Therefore, the value of c is equal to zero.
Hence, the correct option is option (C).