Count ellipse intersections without losing the domain

The curves C1,C2C_1,C_2 have equations (x−2)2=a2(1−y2)(x-2)^2=a^2(1-y^2) and y=x2−4x+1y=\frac{x^2-4}{x+1}, where 1<a<21<a<2, respectively. Describe the geometrical shape of C1C_1. mark

(a) State a sequence of transformations which transforms the graph of x2+y2=1x^2+y^2=1 to the graph of C1C_1. marks

(b) (i) Sketch C1,C2C_1,C_2 on the same diagram, stating the coordinates of any points of intersection with the axes and the equations of any asymptotes. marks

(ii) Show algebraically that the xx-coordinates of the points of intersection of C1,C2C_1,C_2 satisfy the equation (x+1)2(x−2)2=a2(x+1)2−a2(x2−4)2(x+1)^2(x-2)^2=a^2(x+1)^2-a^2(x^2-4)^2. marks

(iii) Deduce the number of real roots of the equation in part (ii).

[2010/HC/Prelim/I/12]

Total marks: 12

Solution

Shape

Recognising a conic

The curves C1C_1 and C2C_2 have equations (x−2)2=a2(1−y2)(x-2)^2=a^2(1-y^2) and y=x2−4x+1y=\dfrac{x^2-4}{x+1} respectively, where 1<a<21<a<2. Describe the geometrical shape of C1C_1.

Hint

  1. Divide by a2a^2 to get (x−2)2a2+y2=1\frac{(x-2)^2}{a^2}+y^2=1.

  2. Compare with the standard ellipse (x−h)2p2+(y−k)2q2=1\frac{(x-h)^2}{p^2}+\frac{(y-k)^2}{q^2}=1.

Answer · Shape

Ellipse centred (2,0)(2,0) with semiaxes aa horizontally and 1 vertically.

(a)

Transforming a circle

State a sequence of transformations which transforms the graph of x2+y2=1x^2+y^2=1 to the graph of C1C_1.

Explore this idea

Hint

  1. Which stretch widens the unit circle to semi-axis aa in the xx-direction only?

  2. Then which translation moves the centre to (2,0)(2,0)?

Answer · (a)

Stretch horizontally by aa, then translate right 2.

Explore the idea

Stretch a circle, then slide it

Start with the unit circle. A horizontal stretch makes an ellipse, and a translation moves its centre to (2, 0).

Teal: x² + y² = 1. No comparison overlay.

Start: x² + y² = 1

The unit circle, centred at the origin.

  • Shape after the selected step
  • Centre of the shape
Ideas to try
  1. Which direction does the ellipse stay the same height in?
  2. Where do the end-points of the horizontal axis end up?
(b)i

Curve sketching

Sketch C1C_1 and C2C_2 on the same diagram, stating the coordinates of any points of intersection with the axes and the equations of any asymptotes.

Explore this idea

Hint

  1. C1C_1: the ellipse crosses the xx-axis at x=2±ax=2\pm a and never meets the yy-axis since a<2a<2.

  2. C2C_2: factorise x2−4x^2-4, find the vertical asymptote and the oblique one by division.

  3. Draw both and look at where they meet.

Q16 ellipse and rational curve
Q16 ellipse and rational curveScroll across to see the full diagram.

Answer · (b)i

C1C_1: (2±a,0)(2\pm a,0). C2C_2: (±2,0),(0,−4)(\pm2,0),(0,-4); asymptotes x=−1,y=x−1x=-1,y=x-1.

1<a<21<a<2; C1 has no y-intercept.

Explore the idea

An ellipse and a rational curve

Switch on each feature of C₂ and the intersections with C₁.

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

  • Ellipse C₁
  • C₂
  • Asymptotes of C₂
  • Two intersections
Ideas to try
  1. Does the ellipse ever meet the left branch of C2C_2?
  2. Which feature stops the ellipse from reaching the yy-axis?
(b)ii

Algebraic substitution

Show algebraically that the xx-coordinates of the points of intersection of C1C_1 and C2C_2 satisfy the equation (x+1)2(x−2)2=a2(x+1)2−a2(x2−4)2(x+1)^2(x-2)^2=a^2(x+1)^2-a^2(x^2-4)^2.

Hint

  1. Substitute y=x2−4x+1y=\frac{x^2-4}{x+1} into (x−2)2=a2(1−y2)(x-2)^2=a^2(1-y^2).

  2. Clear the denominator by multiplying by (x+1)2(x+1)^2.

Answer · (b)ii

(x+1)2(x−2)2=a2(x+1)2−a2(x2−4)2(x+1)^2(x-2)^2=a^2(x+1)^2-a^2(x^2-4)^2.

Substitute C2 into C1 and multiply by (x+1)²; x=−1 is excluded.

(b)iii

Counting roots

Deduce the number of real roots of the equation in part (b)(ii).

Hint

  1. Each real root is the xx-coordinate of an intersection.

  2. Count the intersection points on your sketch.

Answer · (b)iii

2 real roots.