Line-curve intersection conditions

CAIE P1 2020 June Q1

4 marks Standard +0.3

1 Find the set of values of $m$ for which the line with equation $y = m x + 1$ and the curve with equation $y = 3 x ^ { 2 } + 2 x + 4$ intersect at two distinct points.

CAIE P1 2021 March Q4

5 marks Moderate -0.3

4 A line has equation $y = 3 x + k$ and a curve has equation $y = x ^ { 2 } + k x + 6$, where $k$ is a constant. Find the set of values of $k$ for which the line and curve have two distinct points of intersection.

CAIE P1 2022 March Q2

5 marks Moderate -0.3

2 A curve has equation $y = x ^ { 2 } + 2 c x + 4$ and a straight line has equation $y = 4 x + c$, where $c$ is a constant. Find the set of values of $c$ for which the curve and line intersect at two distinct points.

CAIE P1 2020 November Q1

3 marks Moderate -0.3

1 Find the set of values of $m$ for which the line with equation $y = m x - 3$ and the curve with equation $y = 2 x ^ { 2 } + 5$ do not meet.

CAIE P1 2020 November Q4

5 marks Standard +0.3

4 A curve has equation $y = 3 x ^ { 2 } - 4 x + 4$ and a straight line has equation $y = m x + m - 1$, where $m$ is a constant. Find the set of values of $m$ for which the curve and the line have two distinct points of intersection.

CAIE P1 2021 November Q2

5 marks Standard +0.3

2 A curve has equation $y = k x ^ { 2 } + 2 x - k$ and a line has equation $y = k x - 2$, where $k$ is a constant. Find the set of values of $k$ for which the curve and line do not intersect.

CAIE P1 2024 November Q9

10 marks Standard +0.3

9 The equation of a curve is $y = \frac { 1 } { 2 } k ^ { 2 } x ^ { 2 } - 2 k x + 2$ and the equation of a line is $y = k x + p$, where $k$ and $p$ are constants with $0 < k < 1$.

It is given that one of the points of intersection of the curve and the line has coordinates $\left( \frac { 5 } { 2 } , \frac { 1 } { 2 } \right)$. Find the values of $k$ and $p$, and find the coordinates of the other point of intersection. \includegraphics[max width=\textwidth, alt={}, center]{39393c20-34df-4167-a8bf-e4067371de81-15_2725_35_99_20}
It is given instead that the line and the curve do not intersect. Find the set of possible values of $p$.

CAIE P1 2009 June Q2

4 marks Standard +0.3

2 Find the set of values of $k$ for which the line $y = k x - 4$ intersects the curve $y = x ^ { 2 } - 2 x$ at two distinct points.

CAIE P1 2011 June Q2

5 marks Standard +0.3

2 Find the set of values of $m$ for which the line $y = m x + 4$ intersects the curve $y = 3 x ^ { 2 } - 4 x + 7$ at two distinct points.

CAIE P1 2007 November Q1

3 marks Moderate -0.3

1 Determine the set of values of the constant $k$ for which the line $y = 4 x + k$ does not intersect the curve $y = x ^ { 2 }$.

CAIE P1 2009 November Q10

13 marks Standard +0.3

10 \includegraphics[max width=\textwidth, alt={}, center]{b566719c-216e-41e5-8431-da77e1dad73e-4_702_625_260_758}

The diagram shows the line $2 y = x + 5$ and the curve $y = x ^ { 2 } - 4 x + 7$, which intersect at the points $A$ and $B$. Find
(a) the $x$-coordinates of $A$ and $B$,
(b) the equation of the tangent to the curve at $B$,
(c) the acute angle, in degrees correct to 1 decimal place, between this tangent and the line $2 y = x + 5$.
Determine the set of values of $k$ for which the line $2 y = x + k$ does not intersect the curve $y = x ^ { 2 } - 4 x + 7$. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

CAIE P1 2010 November Q6

7 marks Moderate -0.3

6 A curve has equation $y = k x ^ { 2 } + 1$ and a line has equation $y = k x$, where $k$ is a non-zero constant.

Find the set of values of $k$ for which the curve and the line have no common points.
State the value of $k$ for which the line is a tangent to the curve and, for this case, find the coordinates of the point where the line touches the curve.

CAIE P1 2014 November Q5

5 marks Standard +0.3

5 Find the set of values of $k$ for which the line $y = 2 x - k$ meets the curve $y = x ^ { 2 } + k x - 2$ at two distinct points.

CAIE P1 2015 November Q1

3 marks Standard +0.3

1 A line has equation $y = 2 x - 7$ and a curve has equation $y = x ^ { 2 } - 4 x + c$, where $c$ is a constant. Find the set of possible values of $c$ for which the line does not intersect the curve.

CAIE P1 2016 November Q1

3 marks Standard +0.3

1 Find the set of values of $k$ for which the curve $y = k x ^ { 2 } - 3 x$ and the line $y = x - k$ do not meet.

CAIE P1 2017 November Q2

4 marks Standard +0.8

2 Find the set of values of $a$ for which the curve $y = - \frac { 2 } { x }$ and the straight line $y = a x + 3 a$ meet at two distinct points.

CAIE P1 2018 November Q2

4 marks Moderate -0.3

2 A line has equation $y = x + 1$ and a curve has equation $y = x ^ { 2 } + b x + 5$. Find the set of values of the constant $b$ for which the line meets the curve.

CAIE P1 2018 November Q10

9 marks Standard +0.3

10 The equation of a curve is $y = 2 x + \frac { 12 } { x }$ and the equation of a line is $y + x = k$, where $k$ is a constant.

Find the set of values of $k$ for which the line does not meet the curve.
In the case where $k = 15$, the curve intersects the line at points $A$ and $B$.
Find the coordinates of $A$ and $B$.
Find the equation of the perpendicular bisector of the line joining $A$ and $B$.

CAIE P1 2019 November Q6

7 marks Standard +0.3

6 A line has equation $y = 3 k x - 2 k$ and a curve has equation $y = x ^ { 2 } - k x + 2$, where $k$ is a constant.

Find the set of values of $k$ for which the line and curve meet at two distinct points.
For each of two particular values of $k$, the line is a tangent to the curve. Show that these two tangents meet on the $x$-axis.

Edexcel C1 2016 June Q8

8 marks Standard +0.3

8. The straight line with equation $y = 3 x - 7$ does not cross or touch the curve with equation $y = 2 p x ^ { 2 } - 6 p x + 4 p$, where $p$ is a constant.

Show that $4 p ^ { 2 } - 20 p + 9 < 0$
Hence find the set of possible values of $p$.
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Edexcel Paper 2 Specimen Q5

5 marks Standard +0.3

The line $l$ has equation

$$3 x - 2 y = k$$ where $k$ is a real constant.
Given that the line $l$ intersects the curve with equation $$y = 2 x ^ { 2 } - 5$$ at two distinct points, find the range of possible values for $k$.

AQA C1 2015 June Q8

8 marks Moderate -0.3

8 A curve has equation $y = x ^ { 2 } + ( 3 k - 4 ) x + 13$ and a line has equation $y = 2 x + k$, where $k$ is a constant.

Show that the $x$-coordinate of any point of intersection of the line and curve satisfies the equation $$x ^ { 2 } + 3 ( k - 2 ) x + 13 - k = 0$$
Given that the line and the curve do not intersect:
1. show that $9 k ^ { 2 } - 32 k - 16 < 0$;
2. find the possible values of $k$.
  \includegraphics[max width=\textwidth, alt={}]{c7f38f7e-75aa-4b41-96fd-f38f968c225c-18_1657_1714_1050_153}

AQA C1 2008 January Q7

8 marks Moderate -0.3

7 The curve $C$ has equation $y = x ^ { 2 } + 7$. The line $L$ has equation $y = k ( 3 x + 1 )$, where $k$ is a constant.

Show that the $x$-coordinates of any points of intersection of the line $L$ with the curve $C$ satisfy the equation $$x ^ { 2 } - 3 k x + 7 - k = 0$$
The curve $C$ and the line $L$ intersect in two distinct points. Show that $$9 k ^ { 2 } + 4 k - 28 > 0$$
Solve the inequality $9 k ^ { 2 } + 4 k - 28 > 0$.

AQA C1 2009 June Q7

9 marks Standard +0.3

7 The curve $C$ has equation $y = k \left( x ^ { 2 } + 3 \right)$, where $k$ is a constant.
The line $L$ has equation $y = 2 x + 2$.

Show that the $x$-coordinates of any points of intersection of the curve $C$ with the line $L$ satisfy the equation $$k x ^ { 2 } - 2 x + 3 k - 2 = 0$$
The curve $C$ and the line $L$ intersect in two distinct points.
1. Show that $$3 k ^ { 2 } - 2 k - 1 < 0$$
2. Hence find the possible values of $k$.