# 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  