Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2006 | January | Q#2

Question

The point has coordinates and the point has coordinates .

The line has equation .

a.

i.       Show that .

ii.       Hence find the coordinates of the mid-point of .

b.   Find the gradient of .

c.   Line is perpendicular to the line .

i.       Find the gradient of .

ii.       Hence find the equation of the line .

iii.       Given that point lies on the x-axis, find its x-coordinate.

Solution

a.

i.

We are given that point lies on the line whose equation is given as; Since coordinates of a point on the given line must satisfy equation of the lien.

Therefore must satisfy the equation;      ii.

To find the mid-point of a line we must have the coordinates of the end-points of the line.

Expressions for coordinates of mid-point of a line joining points and ;

x-coordinate of mid-point of the line y-coordinate of mid-point of the line For the given case we have the point ( and the point . Therefore;

x-coordinate of mid-point of the line y-coordinate of mid-point of the line Hence mid-point of has coordinates .

b.

Expression for slope (gradient) of a line joining points and ; For the given case we have the point ( and the point . Therefore;   c.

i.

We are given that line is perpendicular to the line .

If two lines are perpendicular (normal) to each other, then product of their slopes and is;  Therefore; From (b) we have ;   ii.

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).

Point-Slope form of the equation of the line is; For line AC we have coordinates one point on the line and also slope from (c:i) as .

Therefore;      iii.

We are given that point lies on x-axis that means it is x-intercept of line 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).

Equation of the line , from (c:ii) is;       