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    Ellipse Questions and Answers

    Uploaded by

    Daniel
    This document provides information about conic sections, specifically ellipses. It defines the standard equations for ellipses, including equations for tangents, normals, foci, and directrices. It then provides 16 multi-part problems involving finding equations of ellipses and their tangents, normals, foci and directrices based on given information, as well as properties of ellipses and relationships between elements of ellipses.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    Ellipse Questions and Answers

    Uploaded by

    Daniel
    1K views7 pages
    This document provides information about conic sections, specifically ellipses. It defines the standard equations for ellipses, including equations for tangents, normals, foci, and directrices. It then provides 16 multi-part problems involving finding equations of ellipses and their tangents, normals, foci and directrices based on given information, as well as properties of ellipses and relationships between elements of ellipses.

    Original Description:

    Ejercicios sobre elipses con respuestas

    Original Title

    Ellipse questions and answers

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    © © All Rights Reserved

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    PDF, TXT or read online from Scribd

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    This document provides information about conic sections, specifically ellipses. It defines the standard equations for ellipses, including equations for tangents, normals, foci, and directrices. It then provides 16 multi-part problems involving finding equations of ellipses and their tangents, normals, foci and directrices based on given information, as well as properties of ellipses and relationships between elements of ellipses.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    1K views7 pages

    Ellipse Questions and Answers

    Uploaded by

    Daniel
    This document provides information about conic sections, specifically ellipses. It defines the standard equations for ellipses, including equations for tangents, normals, foci, and directrices. It then provides 16 multi-part problems involving finding equations of ellipses and their tangents, normals, foci and directrices based on given information, as well as properties of ellipses and relationships between elements of ellipses.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 7

    Further Maths – Conics

    Ellipse
    2 2 �� �� �
    Tangent to 2 + 2 = � �, � � + =

    2 2 ℎ �
    Tangent to 2 + 2 = ℎ, � 2 + 2 =

    2 2
    Normal to 2 + 2 = � �, � � ��
    − �� �
    = −

    2 2 2 2
    Normal to 2 + 2 = ℎ, � ℎ
    − �
    = −

    2
    Foci ± , Directrices = ± = − =√ − 2

    ------------------------------------------------------------------------------------------------------------------------------

    1. Find the foci and directrices for the following ellipses


    2 2 2 2
    a) + = b) + = c) + = .

    2. Find the cartesian equations of the ellipses with the following properties;

    a) Foci at ± , , directrices at = ±9.

    b) Foci at ± , , directrices at = ± .

    3. Find the equations of the tangents and normals at the point given
    2 2
    a) + = , b) + = ,− c) + = ,

    4. Find the equations of the tangent and normal to the ellipse + = at the point

    P(5cost, 2sint).

    5. Find the equations of the tangents to the given ellipses at the specified points;
    � �
    a) x = 4cos�, y= 3sin� at � = . b) x = cos�, y = 2sin� at � = .

    2 2
    6. The tangents at the points , � and , � to the ellipse 2 + 2 = are

    2
    perpendicular. Show that = − 2 .

    7. Show that the line = + is a tangent to the ellipse with equation 9 + = . Find the

    equation of the corresponding normal to the ellipse.


    8. The orbit of the (former) planet Pluto is an ellipse with major axis of length 1.18 x 1010 km. and
    eccentricity . Calculate the length of the minor axis.
    9. The orbit of a satellite is an ellipse of eccentricity with the centre of the Earth as one focus.
    The Earth may be treated as a sphere of radius 6400 km. If the least height of the satellite above
    the earth’s surface is 400 k , what is the greatest height?

    2 2
    10. The point P(acos�, bsin�) is on the ellipse 2 + 2 = . The foci are at the points S and T and
    is the eccentricity of the ellipse. Show that | � − � | = �.

    2 2
    11. Show that if the line = + is tangent to the ellipse 2 + 2 = then + = .

    2 2
    12.* The tangent and normal to the ellipse 2 + 2 = at the point P(acos�, bsin�) meet the x-axis
    at T and G respectively. C is the centre of the ellipse and Y is the foot of the perpendicular from
    C to the tangent at P. Prove that (i) CG.CT = − , (ii) CY.GP = .

    2 2
    13.*The tangent at P(acos�, bsin�) to the ellipse 2 + 2 = cuts the y-axis at Q. The normal at P
    is parallel to the li e joi i g Q to o e focus S’. If S is the other focus, show that PS is parallel to
    the y-axis.

    2 2
    14.* For the ellipse 2 + 2 = a vertical line is drawn through the focus with positive x-coordinate
    which intersects the ellipse at Q where Q has positive y coordinate. The normal at Q passes
    through the lowest point of the minor axis of the ellipse. If is the eccentricity of the ellipse
    show that + − = and hence find to 2 decimal places.

    15.* Let be the perpendicular distance from the origin to the tangent at a point P on the ellipse
    2 2
    2 + 2 = . If � � are the two foci of the ellipse show that
    2
    �� − �� = − 2 .

    2 2
    16.* Find the largest possible distance of any normal to the ellipse 2 + 2 = from the origin
    (excluding the normals at (± , .
    Jan 27­10:47

    Jan 27­10:49

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