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

& y=mx+c \\\ & \Rightarrow 0=m.0+c \\\ & \Rightarrow 0=0+c \\\ & \Rightarrow 0=c \\\ \end{aligned}$$ So, the value of c is zero and also in the equation of a straight line $$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{aligned} & mx=mx+c \\\ & \Rightarrow 0=c \\\ \end{aligned}$$ Therefore, the value of c is equal to zero. Hence, the correct option is option (C).