Line-curve intersection conditions

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.

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.

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.

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.

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.

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.

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.

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.

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

10

1. 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
   1. the \(x\)-coordinates of \(A\) and \(B\),
   2. the equation of the tangent to the curve at \(B\),
   3. the acute angle, in degrees correct to 1 decimal place, between this tangent and the line \(2 y = x + 5\).
   4. Determine the set of values of \(k\) for which the line \(2 y = x + k\) does not intersect the curve \(y = x ^ { 2 } - 4 x + 7\).

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.

1. Find the set of values of \(k\) for which the curve and the line have no common points.
2. 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.

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.

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.

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.

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.

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.

1. Find the set of values of \(k\) for which the line and curve meet at two distinct points.
2. 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.

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.

1. Show that \(4 p ^ { 2 } - 20 p + 9 < 0\)
2. Hence find the set of possible values of \(p\).
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1. 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 \(k\).

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.

1. Show that the \(x\)-coordinate of any point of intersection of the line and curve satisfies the equation $$x ^ { 2 } + 3 ( k - 2 ) x + 13 - k = 0$$
2. 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 \(k\).
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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.

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

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

1. Show that the \(x\)-coordinates of any points of intersection of the curve \(C\) with the line \(L\) satisfy the equation $$k x ^ { 2 } - 2 x + 3 k - 2 = 0$$
2. 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 \(k\).