Enrichment, Extension, and Application

Question 1

1 point

Exercises

  1. Consider a point \( P \) and a plane \( \pi \) with equation \( 3x - 2y + 2z = 0 \). Let \( Q \) be the point of intersection of the plane \( \pi \) and the line through \( P \) with direction vector \( (1, 0, -2) \), and let \( R \) be the point of intersection of the plane \( \pi \) and the line through \( P \) with direction vector \( (1, 1, 0) \).
    1. If the point \( P \) is on the line with equation \( x = y = z \), find all possible positions of \( P \) such that the length of \( QR \) is \( 27 \).
    2. If the point \( P \) is on the line with equation \( x = 0 \), find all possible positions of \( P \) such that the length of \( QR \) is \( 24 \).
    1. Prove that the scalar equation of the plane that cuts the axes at \( A~(a, 0, 0)\), \(B~(0, b, 0) \), and \( C~(0, 0, c) \) is given by \( \dfrac{x}{a} + \dfrac{y}{b} + \dfrac{z}{c} = 1 \).
    2. Calculate the value of the solid bounded by the planes with equations \( 3x + 2y + 5z = 120\), \(5x + 4y + 6z = 120\), \(x = 0\), \(y = 0 \), and \( z = 0 \).
  2. The cube \( ABCDEFGH \) with an edge length of \( 2 \) is sliced with a plane passing through the midpoints of edge \( AE\), \(BC \), and \( GH \), as shown.
    Cube ABCDEFGH of edge 2 and midpoints of AE, BC, GH marked
    Cube ABCDEFGH as described before, with plane passing through the three midpoints described

    Find the shape of the section in the cube and calculate its area.

  3. Given the three lines with parametric equations \( L_1: \begin{cases} x = 1 \\ y = 2 - 2a \\ z = a + 3 \end{cases}, L_2: \begin{cases} x = 17 - 8b \\ y = 6 - 2b \\ z = 9 - 3b \end{cases}, L_3: \begin{cases} x = 1 \\ y = 6c - 4 \\ z = 3c \end{cases} \), where \( a, b, c \) are real numbers, and the plane with equation \( x + 2y - 2z + 7 = 0 \):
    1. Prove that the three lines are concurrent, and find the coordinates of the intersection point, \( V \).
    2. Find the intersection points of the three lines with the plane, denoted \( A, B, C \) respectively for \( L_1, L_2, L_3 \).
    3. Find the volume of tetrahedron \( VABC \).
    1. Show that the set of equations \( \begin{cases} x = 6 - p - t \\ y = p \\ z = t \end{cases}\), \(0 \leq p \leq 2\), \(0 \leq t \leq 3 \) bounds a region whose interior is in the shape of a parallelogram.
    2. A parallelepiped is bordered by the pairs of parallel planes: \begin{align*} P_1: x + 2y + 2z &= 3 & P_2: x + 2y + 2z &= 6 \\ P_3: x - 2y - 2z &= 7 & P_4: x - 2y - 2z &= 19 \\ P_5: x + 2y - 2z &= 11 & P_6: x + 2y - 2z &= 19 \\ \end{align*}
      1. Find the distance between the planes \( P_1 \) and \( P_2 \).
      2. Find the point \( A \) where \( P_1\), \(P_3 \), and \( P_5 \) meet.
      3. Find the volume of the parallelepiped.
    1. Explain why \( x^2 + y^2 + z^2 = 50 \) is the equation of a sphere centred at the origin.
    2. Find the center and the radius of the sphere with equation \( (x - 3)^2 + (y - 5)^2 + (z + 4)^2 = 289 \).
    3. Explain why \( x^2 + y^2 - z^2 = 0 \) is the equation of a cone.
    4. Find the intersection between the sphere in part a) and the cone given in part b). Give both the equation and the interpretation of this figure.
    5. How many points \( P~(a, b, c) \) with \( a, b, c \in \mathbb{Z} \) are situated on the figure found in part d)?
  4. The intersection between the sphere with equation \( (x - 3)^2 + (y - 5)^2 + (z + 4)^2 = 289 \) and the plane with equation \( 6x + 2y - 9z = k \) is a circle of radius \( 15 \).
    1. Find both possible values for \( k \).
    2. For each value of \( k \) found in part a), find the coordinates of the centre of the circle.
  5. Diagram of spherical coordinates as described in the question The Cartesian equation of a sphere centred at the origin \( O \) is \( x^2 + y^2 + z^2 = R^2 \). A set of parametric equations for the same sphere are given by \[ \begin{cases} x = R\cos{( \theta )}\sin{( \phi )} \\ y = R\sin{( \theta )}\sin{( \phi )} \\ z = R\cos{( \phi )} \end{cases} \] where \( \theta \in [0, 360^\circ) \) is the angle that the projection of the line \( OP \) onto the \( xy \)-plane makes with the positive \( x \)-axis (\( \theta = 0^\circ \) corresponds to \( OP \) lying coincident with the positive \( x \)-axis, \( \theta = 180^\circ \) corresponds to \( OP \) lying coincident with the negative \( x \)-axis), and \( \phi \in [0, 180^\circ] \) is the angle that the line \( OP \) makes with the positive \( z \)-axis (\( \phi = 0^\circ \) corresponds to \( OP \) lying coincident to the positive \( z \)-axis, \( \phi = 180^\circ \) corresponds to \( OP \) lying coincident to the negative \( z \)-axis). These equations are also called the spherical equations; \( R\), \(\theta\), and \( \phi \) are called the spherical coordinates of the point \( P \) on the sphere.
    1. Show that any point \( P~\Big( R\cos{(\theta)}\sin{(\phi)}, R\sin{(\theta)}\sin{(\phi)}, R\cos{(\phi)} \Big) \) satisfies the Cartesian equation \( x^2 + y^2 + z^2 = R^2 \).
    2. Find a point \( P \) that has positive integers as coordinates and is situated on the sphere of Cartesian equation \( x^2 + y^2 + z^2 = 49 \). Find the polar coordinates of this point. (Round the angles to the closest degree.)
  6. Consider the following system \[ \begin{cases} x^2 + y^2 + z^2 = 6 \\ z = x^2 + y^2 - 4 \\ x^2 = (y + z)^2 \end{cases}, x, y, z \in \mathbb{R} \]
    1. Give a geometric interpretation of each equation in the system.
    2. Solve the system and interpret the solution.
  7. The intersection between the sphere \( x^2 + y^2 + z^2 = 169 \) and the plane \( 2x + 3y + 6z = 35 \) is a circle, \( \mathcal{C} \). Find the centres of the spheres of radius \( 15 \) that contain the circle \( \mathcal{C} \).
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