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Write the following equations in standard form and identify th
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Write the following equations in standard form and identify the values of \( \mathrm{a}, \mathrm{b} \) and \( \mathrm{c} \). Show your COMPLETE SOLUTION.

  1. \( \mathrm{x}^{2}-9 \mathrm{x}=16 \)
  2. \( 2 y^{2}=-y+5 \)
  3. \( -3 m^{2}=-2 \)
  4. \( 7 x(x-5)=0 \)
  5. \( (y+2)(2 y-1)=0 \)
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Write the standard form of the equation of the hyperbola with the given characteristics. x 2 4 − y 2 50 = 1 Given information: Foci at − 3 6 , 0 , 3 6 , 0 , Vertices at − 2 , 0 , 2 , 0 . Formula used: Standard form of equation of a hyperbola with foci ± c , 0 and vertices ± a , 0 on the x -axis is given by: x 2 a 2 − y 2 b 2 = 1 Standard form of equation of a hyperbola with foci 0 , ± c and vertices 0 , ± a on the y -axis is given by: y 2 a 2 − x 2 b 2 = 1 Focus of hyperbola with vertices as ± a , 0 or 0 , ± a and co- vertices 0 , ± b or ± b , 0 is found as: c 2 = a 2 + b 2 Calculation : In order to write the standard form of the equation of the hyperbola with given foci − 3 6 , 0 , 3 6 , 0 and vertices − 2 , 0 , 2 , 0 first note that both foci and vertices are on x -axis equidistance from the center, so the transverse axis is horizontal and center is the origin. Thus, the equation of the hyperbola will be of the form x 2 a 2 − y 2 b 2 = 1 , with vertices as ± a , 0 and co- vertices 0 , ± b . Since, foci at 3 6 units i.e. c = 3 6 and vertices at 2 units each i.e. a = 2 , so calculate the co-vertices, using the formula c 2 = a 2 + b 2 3 6 2 = 2 2 + b 2 9 × 6 = 4 + b 2 b 2 = 54 − 4 b 2 = 50 b = 50 b = 5 2 Putting, a = 2 and b = 5 2 in the above form of hyperbola: x 2 2 2 − y 2 5 2 2 = 1 x 2 4 − y 2 25 × 2 = 1 x 2 4 − y 2 50 = 1 Thus, the standard form of the equation of hyperbola with given information is: x 2 4 − y 2 50 = 1
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Question Text
Write the following equations in standard form and identify the values of \( \mathrm{a}, \mathrm{b} \) and \( \mathrm{c} \). Show your COMPLETE SOLUTION.
TopicAll Topics
SubjectAlgebra 2
ClassClass 11