# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2016 | June | Q#10

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Question The points P(0, 2) and Q(3, 7) lie on the line , as shown in Figure.

The line is perpendicular to , passes through Q and crosses the x-axis at the point R, as shown in Figure.

Find

a.   an equation for , giving your answer in the form ax + by + c = 0, where a, b and c are integers,

b.   the exact coordinates of R,

c.   the exact area of the quadrilateral ORQP, where O is the origin.

Solution

a.

We are required to find equation of the line .

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 already have coordinates of a point on the line Q(3, 7). Therefore, we need slope of the  normal to write its equation.

We are given that the line is perpendicular to .

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; We can find slope of the line if we have slope of the line .

We need to find the slope of the line .

Expression for slope (gradient) of a line joining points and ; Since line is joined by points P(0, 2) and Q(3, 7), therefore;   Hence;   Now we can write equation of the line .

Point-Slope form of the equation of the line is;         b.

We are required to fins the exact coordinates of point R.

It is evident that we are looking for the coordinates of x-intercept of the 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).

We have found equation of the line in (a) as; We substitute y=0 in this equation;    Hence, exact coordinates of point .

c.

We are required to find the exact area of the quadrilateral ORQP, where O is the origin.

We can spot two triangles in the given quadrilateral ORQP as shown below. It is evident that has and has .

It is evident from the diagram that; Expression for the area of the triangle with base and height is;   Therefore, first we need to find distances OP, OR, QR and QP.

Expression for the distance between two given points and is: We have coordinates of these points as given and found as follows;

 Q(3,7) P(0,2) O(0,0) Hence;    Now we have;    Therefore;        Finally;   