Conic sections 1 Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Conic sections 1 questions. See exactly how to solve problems on conics, parabola, parametric-form, hyperbola.

conicsparabolaparametric-formhyperbolafocusdirectrix
Further Maths70 questionsStep-by-step solutions
Question 1
2 markseasy
The parabola CC has equation y2=12xy^{2}=12x. The point P(3t2, 6t)P\left(3t^{2},\ 6t\right) lies on CC, where tt is a parameter. Find the coordinates of PP when t=2t=2.

Worked solution

  1. Write down the curve and its general parametric point

    y2=12x,P(3t2, 6t)y^{2}=12x,\quad P\left(3t^{2},\ 6t\right)

    The parametric form generates every point of the conic as tt varies.

  2. Evaluate the yy-coordinate

    y=12y=12

    Work out the second parametric coordinate.

  3. State the coordinates of PP

    P(12, 12)P\left(12,\ 12\right)

    These are the required coordinates.

Answer
(12, 12)\left(12,\ 12\right)
Question 2
2 markseasy
The parabola CC has equation y2=8xy^{2}=8x. The point P(2t2, 4t)P\left(2t^{2},\ 4t\right) lies on CC, where tt is a parameter. Find the coordinates of PP when t=1t=-1.

Worked solution

  1. Write down the curve and its general parametric point

    y2=8x,P(2t2, 4t)y^{2}=8x,\quad P\left(2t^{2},\ 4t\right)

    The parametric form generates every point of the conic as tt varies.

  2. Evaluate the yy-coordinate

    y=4y=-4

    Work out the second parametric coordinate.

  3. State the coordinates of PP

    P(2, 4)P\left(2,\ -4\right)

    These are the required coordinates.

Answer
(2, 4)\left(2,\ -4\right)
Question 3
2 markseasy
The rectangular hyperbola HH has equation xy=16xy=16. The point P(4t, 4t)P\left(4t,\ \frac{4}{t}\right) lies on HH, where tt is a non-zero parameter. Find the coordinates of PP when t=2t=2.

Worked solution

  1. Write down the curve and its general parametric point

    xy=16,P(4t, 4t)xy=16,\quad P\left(4t,\ \frac{4}{t}\right)

    The parametric form generates every point of the conic as tt varies.

  2. Evaluate the yy-coordinate

    y=2y=2

    Work out the second parametric coordinate.

  3. State the coordinates of PP

    P(8, 2)P\left(8,\ 2\right)

    These are the required coordinates.

Answer
(8, 2)\left(8,\ 2\right)
Question 4
2 markseasy
The rectangular hyperbola HH has equation xy=36xy=36. The point P(6t, 6t)P\left(6t,\ \frac{6}{t}\right) lies on HH, where tt is a non-zero parameter. Find the coordinates of PP when t=3t=3.

Worked solution

  1. Write down the curve and its general parametric point

    xy=36,P(6t, 6t)xy=36,\quad P\left(6t,\ \frac{6}{t}\right)

    The parametric form generates every point of the conic as tt varies.

  2. Evaluate the yy-coordinate

    y=2y=2

    Work out the second parametric coordinate.

  3. State the coordinates of PP

    P(18, 2)P\left(18,\ 2\right)

    These are the required coordinates.

Answer
(18, 2)\left(18,\ 2\right)
Question 5
2 markseasy
The parabola CC has equation y2=20xy^{2}=20x. The point P(5t2, 10t)P\left(5t^{2},\ 10t\right) lies on CC, where tt is a parameter. Find the coordinates of the focus SS of CC.

Worked solution

  1. Write down the curve and its general parametric point

    y2=20x,P(5t2, 10t)y^{2}=20x,\quad P\left(5t^{2},\ 10t\right)

    The parametric form generates every point of the conic as tt varies.

  2. Recall where the focus lies

    S(a, 0)S\left(a,\ 0\right)

    The focus sits on the axis of symmetry, a distance aa from the vertex.

  3. State the coordinates of the focus

    S(5, 0)S\left(5,\ 0\right)

    Substitute the value of aa.

Answer
(5, 0)\left(5,\ 0\right)

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