# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/11) | Q#5

Question

The equation of a curve is .

i.       Sketch the graph of  for , stating the coordinates of the point of  intersection with the y-axis.

Points P and Q lie on the curve and have x-coordinates of  and  respectively.

ii.       Find the length of PQ correct to 1 decimal place.

The line through P and Q meets the x-axis at H(h,0) and the y-axis at k(0,k).

iii.       Show that and find the value of k.

Solution

i.

Ware required to sketch   for .

We can find the points of the graph as follows.

Now we can sketch required graph from these points as shown below.

Hence coordinates of y-intercept of the curve are (0,2).

ii.

We are given that points P and Q lie on the curve and have x-coordinates of  and  respectively.

We are required to find length of PQ.

Expression to find distance between two given points  and is:

For the given case;

Therefore, we need coordinates of points P and Q.

From the given x-coordinates of both points and equation of the curve, we can find coordinates of  point P and Q as  and .

We can proceed further as;

Hence;

iii.

We are given that x-intercept of the line through P and Q has coordinates  and y-intercept  has coordinates .

We need equation of the line passing through points P and Q to find coordinates of its intercepts.

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 have found in (ii) that coordinates of two points on the line PQ.

Two-Point form of the equation of the line is;

Therefore, equation of line PQ;

Now we can use this equation to find the coordinates of its intercept points.

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 are given that x-intercept of the line has  coordinates. Therefore, substituting these  values in equation of the line;

The point  at which curve (or line) intercepts y-axis, the value of . So we can find the  value of  coordinate by substituting  in the equation of the curve (or line).

Similarly, we can substitute the coordinates of y-intercept of the line  in equation of the line.

Hence