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Conic Section

Long Answer Type

101.

Find: (i) the co-ordinates of foci.
(ii) the co-ordinates of vertices.
(iii) the length of major axis and minor axis.
(iv) the eccentricity.
(v) latus rectum of ellipse.
$4 straight x squared plus 9 straight y squared equals 1$

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102.

Find: (i) the co-ordinates of foci.
(ii) the co-ordinates of vertices.
(iii) the length of major axis and minor axis.
(iv) the eccentricity.
(v) latus rectum of ellipse.
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Short Answer Type

103. Find the equation of the ellipse that satisfies the following condition:
Vertices (± 6, 0), foci (± 4, 0).

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104. Find the equation of the ellipse that satisfies the following condition:
Vertices (0, ± 13), foci (0, ± 5).

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105. Find the equation of an ellipse that satisfies the following conditions
Ends of major axis at

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, ends of minor axis at

left parenthesis 0 comma space plus-or-minus 2 right parenthesis

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106. Find the equation of an ellipse that satisfies the following conditions
Ends of major axis at

space left parenthesis 0 comma space plus-or-minus square root of 13 right parenthesis comma

ends of minor axis at

space space left parenthesis plus-or-minus 2 comma space 0 right parenthesis

. Also find the co-ordinate of foci.

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107. Find the equation of an ellipse that satisfy the following condition:
Length of major axis 26, foci (± 5, 0).

The foci (± 5, 0) are on x-axis.
∴ The equation of ellipse in standard form is $space space straight x squared over straight a squared plus straight y squared over straight b squared equals 1$ ...(i)
Foci are $left parenthesis plus-or-minus 5 comma space 0 right parenthesis space left right arrow space left parenthesis plus-or-minus straight c comma space 0 right parenthesis space rightwards double arrow space space straight c space equals space 5$
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Also, $straight c squared equals straight a squared minus straight b squared space rightwards double arrow space 5 squared equals 13 squared minus straight b squared space rightwards double arrow space 25 space equals space 169 minus straight b squared space rightwards double arrow space straight b squared equals 169 minus 25 space equals space 144$
∴ From (i), equation of ellipse is : $straight x squared over 169 plus straight y squared over 144 equals 1$
Let e be the eccentricity
∴ c = ae $rightwards double arrow$ 5 = 13e $rightwards double arrow$ $straight e equals 15 over 13$