The locus of the vertices of the family of parabolas  from Mat

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 Multiple Choice QuestionsMultiple Choice Questions

21.

For the Hyperbola fraction numerator straight x squared over denominator cos squared space straight alpha end fraction space minus space fraction numerator straight y squared over denominator begin display style sin squared end style begin display style space end style begin display style straight alpha end style end fraction space equals 1 which of the following remains constant when α varies?

  • Eccentricity

  • Directrix

  • Abscissae of vertices

  • Abscissae of vertices

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22.

The equation of a tangent to the parabola y2 = 8x is y = x + 2. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is

  • (−1, 1)

  • (0, 2)

  • (2, 4)

  • (2, 4)

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23.

The locus of the vertices of the family of parabolas straight y space equals fraction numerator straight a cubed straight x squared over denominator 3 end fraction space plus fraction numerator straight a squared straight x over denominator 2 end fraction minus 2 straight a space is

  • xy space equals space 105 over 64
  • xy space equals space 5 over 4
  • xy space equals space 35 over 16
  • xy space equals space 35 over 16


A.

xy space equals space 105 over 64
Parabola colon space straight y space equals fraction numerator straight a cubed straight x squared over denominator 3 end fraction plus fraction numerator straight a squared straight x over denominator 2 end fraction minus 2 straight a
Vertex colon thin space left parenthesis straight alpha comma space straight beta right parenthesis
straight alpha space equals space fraction numerator negative straight a squared divided by 2 over denominator 2 straight a cubed divided by 3 end fraction space equals space minus space fraction numerator 3 over denominator 4 straight a end fraction comma
straight beta space equals space fraction numerator open parentheses begin display style straight a to the power of 4 over 4 end style plus 4. begin display style straight a cubed over 3 end style.2 straight a close parentheses over denominator 4 begin display style straight a cubed over 3 end style end fraction space space equals space fraction numerator open parentheses negative begin display style 1 fourth end style plus begin display style 8 over 3 end style close parentheses straight a to the power of 4 over denominator begin display style 4 over 3 end style straight a cubed end fraction
αβ space equals space minus space fraction numerator 3 over denominator 4 straight a end fraction open parentheses negative 35 over 16 close parentheses straight a space equals space 105 over 64
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24.

In an ellipse, the distance between its foci is 6 and minor axis is 8. Then its eccentricity is

  • 3/5

  • 1/5

  • 2/5

  • 2/5

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25.

If the lines 3x − 4y − 7 = 0 and 2x − 3y − 5 = 0 are two diameters of a circle of area 49π square units, the equation of the circle is

  • x2 + y2 + 2x − 2y − 47 = 0

  • x2 + y2 + 2x − 2y − 62 = 0

  • x2 + y2 − 2x + 2y − 62 = 0

  • x2 + y2 − 2x + 2y − 62 = 0

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26.

Let C be the circle with centre (0, 0) and radius 3 units. The equation of the locus of the mid points of the chords of the circle C that subtend an angle of 2π/3  at its centre is

  • x2+y2 = 3/2

  • x2 + y2 = 1

  • x2+y2 = 27/4

  • x2+y2 = 27/4

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27.

Area of the greatest rectangle that can be inscribed in the ellipse straight x squared over straight a squared plus straight y squared over straight b squared space equals space 1

  • 2ab

  • ab

  • square root of ab
  • square root of ab
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28.

Let P be the point (1, 0) and Q a point on the locus y2 = 8x. The locus of mid point of PQ is

  • y2 – 4x + 2 = 0

  • y2 + 4x + 2 = 0

  • x2 + 4y + 2 =

  • x2 + 4y + 2 =

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29. If space straight omega space equals space fraction numerator straight z over denominator straight z minus begin display style 1 third end style straight i space end fraction space and space vertical line straight omega vertical line space equals 1 comma space then space straight z space lies space on
  • an ellipse

  • a circle

  • a straight line

  • a straight line

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30.

The locus of a point P (α, β) moving under the condition that the line y = αx + β is a tangent to the hyperbola straight x squared over straight a squared space minus straight y squared over straight b squared space equals space 1 space is

  • an ellipse

  • a circle

  • a parabola

  • a parabola

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