Respect both square-root domains

The hyperbola CC passes through the point (2,0)(2,0) and has oblique asymptotes y=−2xy=-2x and y=2xy=2x.

(i) Sketch CC, showing the relevant features of the curve. marks

(ii) Write down the equation of CC. mark

(iii) By adding a suitable curve to your sketch in part (i), solve the inequality x24−1<x−1\sqrt{\frac{x^2}{4}-1}<\sqrt{x-1}. marks

[2019/TJC/Prelim/I/3]

Total marks: 6

Solution

(i)

Sketching a hyperbola

The hyperbola CC passes through the point (2,0)(2,0) and has oblique asymptotes y=−2xy=-2x and y=2xy=2x. Sketch CC, showing the relevant features.

Explore this idea

Hint

  1. The asymptotes cross at the centre. Where is it?

  2. (2,0)(2,0) is on the line of symmetry, so it is a vertex. What is the other vertex?

Q19 horizontal hyperbola and root curve
Q19 horizontal hyperbola and root curveScroll across to see the full diagram.

Answer · (i)

Centre (0,0)(0,0), vertices (±2,0)(\pm2,0), asymptotes y=±2xy=\pm2x.

Explore the idea

A hyperbola opening sideways

Switch on the features.

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

  • Two branches
  • Asymptotes y = ±2x
Ideas to try
Which branch goes through (2,0)(2,0)?
(ii)

Equations of conics

Write down the equation of CC.

Hint

  1. A hyperbola opening sideways with vertices (±2,0)(\pm2,0) has the form x222−y2b2=1\frac{x^2}{2^2}-\frac{y^2}{b^2}=1.

  2. Its asymptotes are y=±b2xy=\pm\frac b2x. Match with ±2x\pm2x.

Answer · (ii)

x2/4−y2/16=1x^2/4-y^2/16=1.

(iii)

Inequalities from graphs

By adding a suitable curve to your sketch in part (i), solve the inequality x24−1<x−1\sqrt{\dfrac{x^2}4-1}<\sqrt{x-1}.

Explore this idea

Hint

  1. Multiply by 4: the left side becomes 4x24−14\sqrt{\frac{x^2}4-1}, the upper half of CC.

  2. The right side becomes 4x−14\sqrt{x-1}: a square-root curve to add.

  3. Both need their own domains: ∣x∣≥2|x|\ge2 and x≥1x\ge1.

Q19 horizontal hyperbola and root curve
Q19 horizontal hyperbola and root curveScroll across to see the full diagram.

Answer · (iii)

2≤x<42\le x<4.

Compare the upper hyperbola with y=4√(x−1); both roots require x≥2.

Explore the idea

Where is the hyperbola below 4√(x − 1)?

Switch on 'Solution' to shade the region between the two curves from x = 2 to x = 4.

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

  • Upper right branch of C
  • Y = 4√(x − 1)
  • Where the hyperbola is below the square-root curve
Ideas to try
  1. Why does the teal curve not exist for x<2x<2?
  2. Check x=3x=3: 9/4−1=1.12<2=1.41\sqrt{9/4-1}=1.12<\sqrt2=1.41.