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Conic Section

Short Answer Type

91. Find equation of parabola for vertex at origin, focus (0, –2), directrix y = 2.

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92. Find equation of parabola for vertex at origin, focus (–3, 0), directrix x = 3.

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93. Find equation of parabola for vertex at origin, axis along the y-axis and passing through the point (3, 2).

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94. Find equation of parabola for vertex at the origin, symmetric with respect to the x-axis and passing through point (2, –3).

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Long Answer Type

95.

The cable of a uniformly loaded suspension bridge hang in the form of a parabola.
The roadway which is horizontal and 100 m long, is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to a roadway 18 m from the middle.

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96. Verify that the locus of a point P which moves so that the sum of its distance from points S₁ (– 4,0) and S₂ (4,0) is 10, is an ellipse.

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97. In the following, find the co-ordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
$straight x squared over 36 plus straight y squared over 16 equals 1$

Equation of ellipse is $straight x squared over 36 plus straight y squared over 16 equals 1$
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∴ Equation of ellipse is $space space straight x squared over straight a squared plus straight y squared over straight b squared equals 1$
Now, $space space space space straight a squared space equals space 36 space rightwards double arrow space space space straight a space equals space 6 space space and space space straight b squared equals 16 space rightwards double arrow space straight b space equals space 4$
Co-ordinates of foci
Focus, $straight S subscript 1 left parenthesis negative straight c comma space 0 right parenthesis space left right arrow space left parenthesis negative square root of straight a squared minus straight b squared end root comma space 0 right parenthesis space left right arrow space left parenthesis negative square root of 36 minus 16 end root comma space 0 right parenthesis space left right arrow space left parenthesis negative square root of 20 comma space 0 right parenthesis space left right arrow space left parenthesis negative 2 square root of 5 comma space 0 right parenthesis$
Focus, $straight S subscript 2 space left parenthesis straight c comma space 0 right parenthesis left right arrow left parenthesis 2 square root of 5 comma space 0 right parenthesis$
Co-ordinates of vertices
$straight A space left parenthesis straight a comma space 0 right parenthesis space left right arrow space left parenthesis 6 comma space 0 right parenthesis$
$straight A apostrophe space left parenthesis negative straight a comma space 0 right parenthesis space left right arrow space left parenthesis negative 6 comma space 0 right parenthesis$
Length of major axis = 2a = 2 (6) = 12
Length of minor axis = 2b = 2 x 4 = 8

Let the eccentricity be e. Now, c = ae $rightwards double arrow$ $2 square root of 5 space equals space 6 straight e space rightwards double arrow space straight e space equals space fraction numerator square root of 5 over denominator 3 end fraction$
∴ $straight e equals fraction numerator square root of 5 over denominator 3 end fraction$
Length of latus rectum, $l equals fraction numerator 2 straight b squared over denominator straight a end fraction space rightwards double arrow space l equals fraction numerator 2 cross times 16 over denominator 6 end fraction equals 16 over 3$

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98. In the following, find the co-ordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.

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99. In the following equation, Find the co-ordinates of foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of latus rectum of the ellipse.

16 straight x squared plus straight y squared equals 16

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100. In the following equation, Find the co-ordinates of foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of latus rectum of the ellipse.

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