Compare a rational branch to the upper hyperbola

The curve CC has equation y=ax+b+2a+2(b+2a)x−2y=ax+b+2a+\frac{2(b+2a)}{x-2}, where a,ba,b are real constants such that a>0,b≠−2a,x≠2a>0,b\ne-2a,x\ne2.

(i) By using differentiation, find the range of values of bb in terms of aa such that CC has no stationary points. marks

(ii) Given that b=−3ab=-3a, sketch CC, stating the equations of any asymptotes and the coordinates of the points where CC crosses the axes. marks

(iii) On the same diagram as CC, sketch the graph of (ya)2−(x−1)2=4(\frac ya)^2-(x-1)^2=4, showing clearly the equations of the asymptotes and turning points. Hence solve the inequality x−1−2x−2≥(x−1)2+4x-1-\frac2{x-2}\ge\sqrt{(x-1)^2+4}. marks

[2018/AJC/Prelim/II/3]

Total marks: 11

Solution

(i)

Stationary points

The curve CC has equation y=ax+b+2a+2(b+2a)x−2y=ax+b+2a+\dfrac{2(b+2a)}{x-2}, where a>0a>0, b≠−2ab\ne-2a and x≠2x\ne2. By differentiation, find the range of values of bb in terms of aa such that CC has no stationary points.

Hint

  1. Differentiate and set dydx=0\frac{dy}{dx}=0.

  2. Isolate (x−2)2(x-2)^2: a square is never negative.

  3. No stationary points means the equation has no real solution.

Answer · (i)

b<−2ab<-2a.

(ii)

Curve sketching

Given that b=−3ab=-3a, sketch CC, stating the equations of any asymptotes and the coordinates of the points where CC crosses the axes.

Explore this idea

Hint

  1. Substitute b=−3ab=-3a and simplify.

  2. Intercepts: x=0x=0 and y=0y=0.

  3. Is the curve increasing? Check dydx\frac{dy}{dx} from (i): b<−2ab<-2a holds.

Q15 rational curve and hyperbola
Q15 rational curve and hyperbolaScroll across to see the full diagram.

Answer · (ii)

Intercepts (0,0),(3,0)(0,0),(3,0); asymptotes x=2,y=a(x−1)x=2,y=a(x-1); increasing on each branch.

a>0,b=−3aa>0,b=-3a.

Explore the idea

Two rising branches

y = x − 1 − 2/(x − 2), with b = −3a and a = 1.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

  • C
  • Oblique asymptote y = a(x − 1)
  • Axis intercepts
Ideas to try
Why does the left branch never turn around?
(iii)a

Hyperbolas

On the same diagram as CC, sketch the graph of (ya)2−(x−1)2=4\left(\dfrac ya\right)^2-(x-1)^2=4, showing clearly the equations of the asymptotes and the turning points.

Explore this idea

Hint

  1. Write it as (y2a)2−(x−12)2=1\left(\frac{y}{2a}\right)^2-\left(\frac{x-1}{2}\right)^2=1: a hyperbola opening up and down.

  2. Asymptotes: set the right side to 0. Vertices: put x=1x=1.

Q15 rational curve and hyperbola
Q15 rational curve and hyperbolaScroll across to see the full diagram.

Answer · (iii)a

Vertices (1,±2a)(1,\pm2a); asymptotes y=±a(x−1)y=\pm a(x-1).

Explore the idea

The hyperbola on the same axes

Its upper branch is y = a√((x − 1)² + 4). One asymptote is shared with C.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

  • C
  • Two branches of the hyperbola
  • Hyperbola's asymptotes y = ±a(x − 1)
Ideas to try
  1. Which asymptote does C share with the hyperbola?
  2. Does C ever cross the upper branch?
(iii)b

Inequalities from graphs

Hence solve the inequality x−1−2x−2≥(x−1)2+4x-1-\dfrac{2}{x-2}\ge\sqrt{(x-1)^2+4}.

Explore this idea

Hint

  1. Multiply both sides by aa: the left side becomes the curve CC and the right side becomes the upper branch of the hyperbola.

  2. Find where CC lies on or above the upper branch.

  3. Remember the domain: x≠2x\ne2.

Answer · (iii)b

1≤x<21\le x<2.

Explore the idea

Where C is on or above the upper hyperbola

Switch on 'Solution' to shade the region between the upper branch and C, from x = 1 to the asymptote x = 2.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

  • C
  • Upper branch y = a√((x−1)²+4)
  • Where C ≥ upper branch: the solution
Ideas to try
  1. At which point do the curves meet, and why is it the left end of the interval?
  2. What happens to C just to the right of x=2x=2?