Worked solution
Use an asymptote family to find when no real roots occur
C has equation .
(i) Find the asymptotes. mark
(ii) Sketch C, labelling asymptotes, axial intercepts and turning points. marks
(iii) Hence find the values of k for which has no real roots. marks
[2013/YJC/I/12]
CJC · Discussion · Question 2
Answer
Asymptotes x=2,y=x+2; curve turns (0,0),(4,8); no-real-root range 0<k≤1.
(i) Divide the numerator by the denominator
, giving asymptotes x=2 and y=x+2.
(ii) Label the two branches and stationary points
The only axial intercept is (0,0), a local maximum on the left branch. The right branch has a local minimum (4,8). These follow from . The source sketch also labels the oblique line’s axis crossings (−2,0) and (0,2); they are not extra curve intercepts.
(iii) Compare with lines through a fixed point
becomes for x≠2. The source compares C with y=k(x+2), which passes through (−2,0)for every k. No roots means no intersections, giving 0<k≤1. No root is lost by excluding x=2, since x=2 never satisfies the original equation.
(iii) Check the endpoint algebraically
The original equation is . For 0<k<1 its required x²is negative. At k=1 it becomes 0=−4, still impossible. At k=0 there is a real root x=0; outside 0<k≤1 the required x²is nonnegative. This explains the source’s included endpoint 1.
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Common mistakes
The asymptote’s axis crossings are not curve intercepts. Include k=1 even though the quadratic degenerates.
Solution notes
At k=1 the equation becomes 0=−4, so this endpoint belongs to the no-real-root range.
The oblique line’s axis crossings are not additional intercepts of the curve.