A family of lines can miss a rational curve

The curve CC has equation y=x2+bx+ay=\frac{x^2+b}{x+a}, where a>0,b>0a>0,b>0.

(i) State the coordinates of the intersection(s) of the curve CC with the axes in terms of a,ba,b. mark

(ii) Find the equation(s) of the asymptote(s). marks

(iii) Draw a sketch of the curve CC, labelling the equation(s) of its asymptote(s) and coordinates of any intersection with the axes. marks

(iv) Hence find the range of values of kk, where kk is a positive constant, for which the equation x2+b=(x+a)(kx−a)x^2+b=(x+a)(kx-a) has no real root. marks

[2009/SRJC/Prelim/I/7]

Total marks: 7

Solution

(i)

Intercepts

The curve CC has equation y=x2+bx+ay=\dfrac{x^2+b}{x+a}, where a>0a>0 and b>0b>0. State the coordinates of the intersection(s) of CC with the axes in terms of aa and bb.

Hint

  1. xx-intercept: y=0y=0. yy-intercept: x=0x=0.

Answer · (i)

Only (0,b/a)(0,b/a); no x-intercept.

(ii)

Asymptotes

Find the equation(s) of the asymptote(s) of CC.

Hint

  1. Vertical: zero of the denominator.

  2. Oblique: divide the numerator by the denominator and read the quotient.

Answer · (ii)

x=−a, y=x−ax=-a,\ y=x-a.

(iii)

Curve sketching

Draw a sketch of CC, labelling the equation(s) of its asymptote(s) and the coordinates of any intersection with the axes.

Explore this idea

Hint

  1. Draw the two asymptotes first.

  2. Which side of each does the curve lie? Look at the sign of a2+bx+a\frac{a^2+b}{x+a}.

  3. Place the intercept (0,ba)\left(0,\frac ba\right).

Q12 source rational sketch
Q12 source rational sketchScroll across to see the full diagram.

Answer · (iii)

Negative branch for x<−ax<-a, positive branch for x>−ax>-a, with the stated intercept/asymptotes.

a,b>0a,b>0; representative parameter sketch.

Explore the idea

A hyperbola with a slanted asymptote

Drawn with a = b = 1: y = (x² + 1)/(x + 1) = (x − 1) + 2/(x + 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: y = (x² + b)/(x + a)
  • Oblique asymptote y = x − a
  • Y-intercept
Ideas to try
  1. Which branch is above the slanted asymptote? Which is below?
  2. Where would the curve cross the xx-axis if bb were negative?
(iv)

Roots from a graph

Hence find the range of values of kk, where kk is a positive constant, for which the equation x2+b=(x+a)(kx−a)x^2+b=(x+a)(kx-a) has no real root.

Explore this idea

Hint

  1. Divide by x+ax+a to compare the curve with the line y=kx−ay=kx-a.

  2. The line passes through (0,−a)(0,-a) and has gradient kk.

  3. Which gradients give a line that never meets the curve?

Answer · (iv)

0<k≤10<k\le1.

Explore the idea

Which gradients miss the curve?

The line passes through (0, −1). Move k and watch how many times it meets the curve.

Gradient of the linek = 0.5
Try an example
Test boundary gradients

Buttons use full-precision boundary values; numbers shown here are rounded.

Number of distinct intersections
0

No intersection. For the positive gradients required in the question, this gives 0<k≤10<k\le1.

There are no isolated intersections at this value. A touching point counts once; hollow endpoints are excluded.

  • Y = (x² + 1)/(x + 1)
  • Y = kx − 1, always through (0, −1)
  • Intersections: the real roots
  • Intersection selected in the coordinate table
Key idea

The line pivots about (0,−a)(0,-a). It misses CC exactly when it is no steeper than the oblique asymptote, which gives 0<k≤10<k\le1.

Ideas to try
  1. Slide k from 0 to 2.5. At exactly which value does the line start to meet the curve?
  2. What happens when the line coincides with the oblique asymptote?