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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.

104 Views

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.

91 Views

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

94 Views

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

148 Views

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

165 Views

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.

181 Views

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.

Let OAB be the triangle with side p.
In $straight capital delta space OLA comma space OA space equals space straight p$
Now,    $OL over straight p space equals space cos space 30 degree space equals space fraction numerator square root of 3 over denominator 2 end fraction space rightwards double arrow space OL space equals space fraction numerator square root of 3 over denominator 2 end fraction straight p$
and    $AL over straight p space equals space sin space 30 degree space equals space 1 half space rightwards double arrow space AL space equals space 1 half straight p$
∴ Point A is $open parentheses fraction numerator square root of 3 over denominator 2 end fraction straight p comma space 1 half straight p close parentheses$
Since it lies on $straight y squared equals 4 ax$
∴ $open parentheses 1 half straight p close parentheses squared space equals space 4 straight a open parentheses fraction numerator square root of 3 over denominator 2 end fraction straight p close parentheses space rightwards double arrow space 1 fourth straight p squared space equals space 2 straight a square root of 3 straight p space rightwards double arrow space straight p equals 8 square root of 3 straight a$
Hence, (i) each side of $space space space space space space ΔABC space equals straight p equals space 8 square root of 3 straight a$ , (ii) area of $ΔABC space equals space fraction numerator square root of 3 straight p squared over denominator 4 end fraction equals fraction numerator square root of 3 over denominator 4 end fraction left parenthesis 64 cross times 3 straight a squared right parenthesis space equals space 48 square root of 3 straight a squared$