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Conic Section

Short Answer Type

51. In the following, find the co-ordinates of the focus, axis of the parabola, the equation of the directrix and length of the latus rectum.

space space space space straight x squared equals 6 straight y

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52. In the following, find the co-ordinates of the focus, axis of the parabola, the equation of the directrix and length of the latus rectum.

straight x squared space equals negative 9 straight y

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53. Find: the co-ordinates of the focus and the vertex, the equations of axis, tangent at the vertex and directrix, the length of latus rectum of the parabola.

2 straight y squared minus 9 straight x equals 0

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54. Find: the co-ordinates of the focus and the vertex, the equations of axis, tangent at the vertex and directrix, the length of latus rectum of the parabola.

4 straight x squared plus 9 straight y equals 0

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55. Find the equation of the parabola whose focus is at point S (2, 5) and the directrix is line 3x + 4y + 1 = 0.

Let line l be the directrix with equation
3x + 4y + 1 = 0 and S (2, 5) be the focus.
Take a point $straight P space left parenthesis straight alpha comma space straight beta right parenthesis$ on the parabola
∴ By definition, PS = 'PM    $rightwards double arrow$ d = p
$rightwards double arrow$ $square root of left parenthesis straight alpha minus 2 right parenthesis squared plus left parenthesis straight beta minus 5 right parenthesis squared end root space equals space open vertical bar fraction numerator 3 straight alpha plus 4 straight beta plus 1 over denominator square root of left parenthesis 3 right parenthesis squared plus left parenthesis 4 right parenthesis squared end root end fraction close vertical bar$
$rightwards double arrow$ $left parenthesis straight alpha minus 2 right parenthesis squared plus left parenthesis straight beta minus 5 right parenthesis squared space equals space fraction numerator left parenthesis 3 straight alpha plus 4 straight beta plus 1 right parenthesis squared over denominator 25 end fraction$
$rightwards double arrow$    $25 straight alpha squared plus 25 straight beta squared minus 100 straight alpha minus 250 straight beta plus 725 equals 9 straight alpha squared plus 16 straight beta squared plus 24 αβ plus 6 straight alpha plus 8 straight beta plus 1$
$rightwards double arrow$    $space space space space space space 16 straight alpha squared plus 9 straight beta squared minus 24 αβ minus 106 straight alpha minus 258 straight beta plus 724 equals 0$
Hence, the locus of P or the equation of parabola is :
$16 straight x squared plus 9 straight y squared minus 24 xy minus 106 straight x minus 258 straight y plus 724 equals 0$

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56. Find the equation of the parabola whose vertex is at (0, 0) and the focus is at point S (5, 0).

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57. Find the equation of the parabola whose vertex is at (0, 0) and the focus is at point (-3, 0).

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58. Find the equation of the parabola whose vertex is at (0, 0) and focus is at (0, – 2).

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59. Find the equation of the parabola that satisfying the following condition:
Vertex (0,0) passing through (2,3) and axis is along x-axis.

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60. Find the equation of the parabola that satisfying the following condition:
Vertex at (0,0), symmetrical about y-axis and passing through the point (2, –3)

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