Final answer:
The statement in question is false because the gradient of a line and the gradient of a perpendicular line are negative reciprocals of each other, not reciprocals.
Step-by-step explanation:
The student's question pertains to the relationship between the gradient of a given line L and the gradient of a line perpendicular to it. The question asks for the validation of the statement that the gradient of line L, which passes through points A(2x,4) and B(-1,x), is equal to the reciprocal of the gradient of the line perpendicular to L passing through point B.
To solve this, we first calculate the gradient (slope) of line L.
The slope of line L is given by the formula: slope of L = (y2 - y1)/(x2 - x1) where (x1, y1) and (x2, y2) are the coordinates of two points on the line. For line L, using points A(2x,4) and B(-1,x), this becomes:
slope of L = (x - 4)/(-1 - 2x)
The slope of a line perpendicular to line L, which we'll call line M, is the negative reciprocal of the slope of line L.
Therefore, if the slope of line L is 'm', the slope of line M will be '-1/m'.
Based on this, the statement in question is false as it incorrectly asserts the slope of line L to be the reciprocal of the slope of line M instead of the negative reciprocal.