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Conic Section

Short Answer Type

61. Find the equation of the parabola that satisfying the following condition:
Vertex at (0,0), focus on the positive x-axis and length of latus rectum

16 over 9.

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62. Find the equation of the parabola that satisfying the following condition:
Vertex at (0, 0) focus on the negative side of y-axis and latus rectum equal to it.

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63. Find the equation of a parabola that satisfies the given condition:
Focus (6, 0), directrix is x = – 6

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64. Find the equation of a parabola that satisfies the given condition:
Focus (0 – 3); directrix y = 3

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65. Find the co-ordinates of a point on the parabola y² = l8x, where the ordinate is 3 times the abscissa.

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66. Find the area of the triangle formed by the lines joining the vertex of the parabola x² = 12y to the ends of the latus rectum.

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67.

An equilateral triangle is inscribed in the parabola $straight y squared equals 4 ax.$ , where one vertex is at the vertex of the parabola. Find (a) the length of the side of the triangle, (b) area of triangle ABC.

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68. If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.

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69. LL' is the latus rectum of a parabola, y² = 4ax, a > 0. Find the co-ordinates of points L and L'.

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70. LL' is the latus rectum of a parabola, x² = -8y. Find the co-ordinates of points L and L'.

Equation of parabola is $straight x squared equals negative 8 straight y$
$rightwards double arrow$ 4a = 8 $rightwards double arrow$ a = 2 $rightwards double arrow$ Focus is at point (0, -2).
$rightwards double arrow$ OS = -2
Let the co-ordinates of L be $left parenthesis straight alpha comma space minus 2 right parenthesis$ and of L' be $left parenthesis negative straight alpha comma space minus 2 right parenthesis space with space straight alpha greater than 0$
Since L $left parenthesis straight alpha comma space minus 2 right parenthesis$ lies on $straight x squared equals negative 8 straight y$
$rightwards double arrow$ $straight alpha squared space equals negative 8 space left parenthesis negative 2 right parenthesis space equals space 16 space rightwards double arrow space straight alpha space equals space 4$
∴ Coordinates of L are (4, -2) and of L' are (-4, -2)