Line intersection: unknown constant then intersect

Questions where one or more lines contain an unknown constant, requiring the student to first find the constant given that the lines intersect, then possibly find the intersection point.

6 questions · Standard +0.3

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Edexcel C4 2009 January Q17
Standard +0.3
17 \end{array} \right) + \lambda \left( \begin{array} { c } - 2
1
- 4 \end{array} \right) \quad l _ { 2 } : \quad \mathbf { r } = \left( \begin{array} { c } - 5
11
p \end{array} \right) + \mu \left( \begin{array} { l } q
2
2 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are parameters and \(p\) and \(q\) are constants. Given that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular,
  1. show that \(q = - 3\). Given further that \(l _ { 1 }\) and \(l _ { 2 }\) intersect, find
  2. the value of \(p\),
  3. the coordinates of the point of intersection. The point \(A\) lies on \(l _ { 1 }\) and has position vector \(\left( \begin{array} { c } 9 \\ 3 \\ 13 \end{array} \right)\). The point \(C\) lies on \(l _ { 2 }\).\\ Given that a circle, with centre \(C\), cuts the line \(l _ { 1 }\) at the points \(A\) and \(B\),
  4. find the position vector of \(B\).\\ 5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a5579938-e202-4543-8513-6483ede49850-09_696_686_196_626} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A container is made in the shape of a hollow inverted right circular cone. The height of the container is 24 cm and the radius is 16 cm , as shown in Figure 2. Water is flowing into the container. When the height of water is \(h \mathrm {~cm}\), the surface of the water has radius \(r \mathrm {~cm}\) and the volume of water is \(V \mathrm {~cm} ^ { 3 }\).
  5. Show that \(V = \frac { 4 \pi h ^ { 3 } } { 27 }\).\\[0pt] [The volume \(V\) of a right circular cone with vertical height \(h\) and base radius \(r\) is given by the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\).] Water flows into the container at a rate of \(8 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\).
  6. Find, in terms of \(\pi\), the rate of change of \(h\) when \(h = 12\). 6. (a) Find \(\int \tan ^ { 2 } x \mathrm {~d} x\).
  7. Use integration by parts to find \(\int \frac { 1 } { x ^ { 3 } } \ln x \mathrm {~d} x\).
  8. Use the substitution \(u = 1 + e ^ { x }\) to show that $$\int \frac { \mathrm { e } ^ { 3 x } } { 1 + \mathrm { e } ^ { x } } \mathrm {~d} x = \frac { 1 } { 2 } \mathrm { e } ^ { 2 x } - \mathrm { e } ^ { x } + \ln \left( 1 + \mathrm { e } ^ { x } \right) + k$$ where \(k\) is a constant.\\ 7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a5579938-e202-4543-8513-6483ede49850-13_511_714_237_612} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The curve \(C\) shown in Figure 3 has parametric equations $$x = t ^ { 3 } - 8 t , \quad y = t ^ { 2 }$$ where \(t\) is a parameter. Given that the point \(A\) has parameter \(t = - 1\),
  9. find the coordinates of \(A\). The line \(l\) is the tangent to \(C\) at \(A\).
  10. Show that an equation for \(l\) is \(2 x - 5 y - 9 = 0\). The line \(l\) also intersects the curve at the point \(B\).
  11. Find the coordinates of \(B\).
Edexcel C4 2014 June Q6
10 marks Standard +0.2
6. With respect to a fixed origin, the point \(A\) with position vector \(\mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\) lies on the line \(l _ { 1 }\) with equation $$\mathbf { r } = \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { r } 0 \\ 2 \\ - 1 \end{array} \right) , \quad \text { where } \lambda \text { is a scalar parameter, }$$ and the point \(B\) with position vector \(4 \mathbf { i } + p \mathbf { j } + 3 \mathbf { k }\), where \(p\) is a constant, lies on the line \(l _ { 2 }\) with equation $$\mathbf { r } = \left( \begin{array} { l } 7 \\ 0 \\ 7 \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ - 5 \\ 4 \end{array} \right) , \quad \text { where } \mu \text { is a scalar parameter. }$$
  1. Find the value of the constant \(p\).
  2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the position vector of their point of intersection, \(C\).
  3. Find the size of the angle \(A C B\), giving your answer in degrees to 3 significant figures.
  4. Find the area of the triangle \(A B C\), giving your answer to 3 significant figures. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 6 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
OCR C4 2006 January Q9
13 marks Standard +0.3
9 Two lines have vector equations $$\mathbf { r } = \left( \begin{array} { r } 4 \\ 2 \\ - 6 \end{array} \right) + t \left( \begin{array} { r } - 8 \\ 1 \\ - 2 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } - 2 \\ a \\ - 2 \end{array} \right) + s \left( \begin{array} { r } - 9 \\ 2 \\ - 5 \end{array} \right) ,$$ where \(a\) is a constant.
  1. Calculate the acute angle between the lines.
  2. Given that these two lines intersect, find \(a\) and the point of intersection.
OCR C4 Q7
10 marks Standard +0.3
7. Relative to a fixed origin, two lines have the equations
and $$\begin{aligned} & \mathbf { r } = \left( \begin{array} { c } 7 \\ 0 \\ - 3 \end{array} \right) + s \left( \begin{array} { c } 5 \\ 4 \\ - 2 \end{array} \right) \\ & \mathbf { r } = \left( \begin{array} { l } a \\ 6 \\ 3 \end{array} \right) + t \left( \begin{array} { c } - 5 \\ 14 \\ 2 \end{array} \right) , \end{aligned}$$ where \(a\) is a constant and \(s\) and \(t\) are scalar parameters.
Given that the two lines intersect,
  1. find the position vector of their point of intersection,
  2. find the value of \(a\). Given also that \(\theta\) is the acute angle between the lines,
  3. find the value of \(\cos \theta\) in the form \(k \sqrt { 5 }\) where \(k\) is rational.
Edexcel C4 Q4
9 marks Standard +0.3
4. The line \(l _ { 1 }\) passes through the points \(P\) and \(Q\) with position vectors ( \(- \mathbf { i } - 8 \mathbf { j } + 3 \mathbf { k }\) ) and ( \(2 \mathbf { i } - 9 \mathbf { j } + \mathbf { k }\) ) respectively, relative to a fixed origin.
  1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has the equation $$\mathbf { r } = ( 6 \mathbf { i } + a \mathbf { j } + b \mathbf { k } ) + \mu ( \mathbf { i } + 4 \mathbf { j } - \mathbf { k } )$$ and also passes through the point \(Q\).
  2. Find the values of the constants \(a\) and \(b\).
  3. Find, in degrees to 1 decimal place, the acute angle between lines \(l _ { 1 }\) and \(l _ { 2 }\).
    4. continued
AQA Further AS Paper 1 2022 June Q7
9 marks Standard +0.3
7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left[ \begin{array} { c } 3 \\ 1 \\ - 2 \end{array} \right] + \lambda \left[ \begin{array} { c } 3 \\ - 4 \\ 1 \end{array} \right] \\ & l _ { 2 } : \mathbf { r } = \left[ \begin{array} { c } - 12 \\ a \\ - 3 \end{array} \right] + \mu \left[ \begin{array} { c } 3 \\ 2 \\ - 1 \end{array} \right] \end{aligned}$$ 7
  1. Show that the point \(P ( - 3,9 , - 4 )\) lies on \(l _ { 1 }\) 7
  2. Show that \(l _ { 1 }\) is perpendicular to \(l _ { 2 }\) 7
  3. Given that the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect, calculate the value of the constant \(a\) 7
  4. Hence, find the coordinates of the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\)