Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2011 | January | Q#9
The line L1 has equation 2y − 3x − k = 0, where k is a constant.
Given that the point A (1, 4) lies on L1, find
a. the value of k,
b. the gradient of L1.
The line L2 passes through A and is perpendicular to L1.
c. Find an equation of L2 giving your answer in the form ax + by + c = 0, where a, b and c are integers.
The line L2 crosses the x-axis at the point B.
d. Find the coordinates of B.
e. Find the exact length of AB.
We are given that line L1 has equation;
We are given that the point A (1, 4) lies on L1.
If a point P(x,y) lies on a line, then its coordinates satisfy the equation of the line.
Therefore, we substitute coordinates of point A (1,4) in equation of L1.
We are required to find the gradient of line L1.
Slope-Intercept form of the equation of the line;
Where is the slope of the line.
We can rearrange the equation of the line in slope-intercept for to find the gradient.
From (a) we have found that k=5;
Hence, gradient of line L1 is;
We are required to find equation of L2.
To find the equation of the line either we need coordinates of the two points on the line (Two-Point form of Equation of Line) or coordinates of one point on the line and slope of the line (Point-Slope form of Equation of Line).
We are given that line L2 passes through the point A(1,4) and is perpendicular to line L1.
If a line is normal to the curve , then product of their slopes and at that point (where line is normal to the curve) is;
Therefore, if we have slope of line L1 we can find slope of line L2.
From (a) we have found that;
Now we can write the equation of line L2.
Point-Slope form of the equation of the line is;
We are given that line L2 crosses x-axis at point B which means point B is xintercept of line L2.
We are required to find the coordinates of x-intercept of line L2.
The point at which curve (or line) intercepts x-axis, the value of . So we can find the value of coordinate by substituting in the equation of the curve (or line).
Hence, coordinates of point B are (7,0).
We are required to find the exact length of AB.
Expression to find distance between two given points and is:
We are given coordinates of point A as (1,4) and we have found coordinates of point B in (d) as (7,0).