4PM1/02R Jan 2020, Question 11: When a Line Meets, Touches or Misses a Curve

Part (a) · 2 marks

State the equations of the two asymptotes of C, one parallel to the x-axis and one parallel to the y-axis.

1. 1As x → ±∞, y → 3: y = 3
2. 2Denominator zero at x = −1: x = −1

Answer(i) y = 3 (ii) x = −1

Part (b) · 2 marks

Find the coordinates where C meets the x-axis and where it meets the y-axis.

1. 1y = 0: 3x − 2 = 0, x = 2/3
2. 2x = 0: y = −2/1 = −2

Answer(i) (2/3, 0) (ii) (0, −2)

Part (c) · 3 marks

Draw the curve C, clearly marking both asymptotes and the coordinates of its axis crossings.

1. 1y = 3 − 5/(x + 1)
2. 2Draw asymptotes x = −1 and y = 3
3. 3Right branch rises from below through (0, −2) and (2/3, 0) towards y = 3
4. 4Left branch lies above y = 3, to the left of x = −1

AnswerTwo-branch hyperbola with asymptotes x = −1 and y = 3; one branch top-left of (−1, 3), the other bottom-right passing through (0, −2) and (2/3, 0)

Part (d) · 7 marks

Given line l with y = mx + 4 misses C, show algebraically that m lies between a − 2√b and a + 2√b, finding integers a and b.

1. 1(3x − 2)/(x + 1) = mx + 4
2. 23x − 2 = mx² + (m + 4)x + 4, so mx² + (m + 1)x + 6 = 0
3. 3No intersections: b² − 4ac < 0, (m + 1)² − 24m < 0
4. 4m² − 22m + 1 < 0
5. 5Roots m = 11 ± √120 = 11 ± 2√30
6. 611 − 2√30 < m < 11 + 2√30

Answer11 − 2√30 < m < 11 + 2√30 (a = 11, b = 30)

More past-paper questions on when a line meets, touches or misses a curve

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