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2.   An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

Since the Axis of the parabola is vertical, Let the equation of the parabola be,  it can be seen that this curve will pass through the point (5/2, 10) if we assume origin at the bottom end of the parabolic arch. So, Hence, the equation of the parabola is  Now, when y = 2 the value of x will be  Hence the width of the arch at this height is 

15. Find the equations of the hyperbola satisfying the given conditions.

       Foci (0,\pm\sqrt{10}), passing through (2,3)

Given, in a hyperbola,  Foci , passing through (2,3) Since foci of the hyperbola are in Y-axis, the equation of the hyperbola will be of the form ; By comparing standard parameter (foci) with the given one, we get Now As we know, in a hyperbola  Now As the hyperbola passes through the point (2,3) Solving Equation (1) and (2) Now, as we know that in a hyperbola  is always greater...

14.  Find the equations of the hyperbola satisfying the given conditions.

      vertices (± 7,0), e = \frac{4}{3}

Given, in a hyperbola vertices (± 7,0), And   Here, Vertices is  on the X-axis so, the standard equation of the Hyperbola will be ; By comparing the standard parameter (Vertices and eccentricity) with the given one, we get  and  From here, Now, As we know the relation  in a hyperbola  Hence, The Equation of the hyperbola is ;

13. Find the equations of the hyperbola satisfying the given conditions.

      Foci (± 4, 0), the latus rectum is of length 12

Given, in a hyperbola Foci (± 4, 0), the latus rectum is of length 12 Here,  focii are on the X-axis so, the standard equation of the Hyperbola will be ; By comparing standard parameter (length of latus rectum and foci) with the given one, we get  and  Now, As we know the relation  in a hyperbola  Since  can never be negative, Hence, The Equation of the hyperbola is ;

12.  Find the equations of the hyperbola satisfying the given conditions.

       Foci (\pm 3\sqrt5, 0), the latus rectum is of length 8.

Given, in a hyperbola Foci , the latus rectum is of length 8. Here,  focii are on the X-axis so, the standard equation of the Hyperbola will be ; By comparing standard parameter (length of latus rectum and foci) with the given one, we get  and  Now, As we know the relation  in a hyperbola  Since  can never be negative, Hence, The Equation of the hyperbola is ;

11.  Find the equations of the hyperbola satisfying the given conditions.

       Foci (0, ±13), the conjugate axis is of length 24.

Given, in a hyperbola Foci (0, ±13), the conjugate axis is of length 24. Here, focii are on the Y-axis so, the standard equation of the Hyperbola will be ; By comparing the standard parameter (length of conjugate axis and foci) with the given one, we get  and  Now, As we know the relation  in a hyperbola  Hence, The Equation of the hyperbola is ; .

10. Find the equations of the hyperbola satisfying the given conditions.

      Foci (± 5, 0), the transverse axis is of length 8.

Given, in a hyperbola Foci (± 5, 0), the transverse axis is of length 8. Here,  focii are on the X-axis so, the standard equation of the Hyperbola will be ; By comparing the standard parameter (transverse axis length and foci) with the given one, we get  and  Now, As we know the relation  in a hyperbola  Hence, The Equation of the hyperbola is ;

9.  Find the equations of the hyperbola satisfying the given conditions.

    Vertices (0, ± 3), foci (0, ± 5)

Given, in a hyperbola  Vertices (0, ± 3), foci (0, ± 5) Here, Vertices and focii are on the Y-axis so, the standard equation of the Hyperbola will be ; By comparing the standard parameter (Vertices and foci) with the given one, we get  and  Now, As we know the relation  in a hyperbola  Hence, The Equation of the hyperbola is ; .

8.  Find the equations of the hyperbola satisfying the given conditions.

       Vertices (0, ± 5), foci (0, ± 8)

Given, in a hyperbola Vertices (0, ± 5), foci (0, ± 8) Here, Vertices and focii are on the Y-axis so, the standard equation of the Hyperbola will be ; By comparing the standard parameter (Vertices and foci) with the given one, we get  and  Now, As we know the relation  in a hyperbola  Hence, The Equation of the hyperbola is ; .

7. Find the equations of the hyperbola satisfying the given conditions. 

       Vertices (± 2, 0), foci (± 3, 0)

Given, in a hyperbola Vertices (± 2, 0), foci (± 3, 0) Here, Vertices and focii are on the X-axis so, the standard equation of the Hyperbola will be ; By comparing the standard parameter (Vertices and foci) with the given one, we get  and  Now, As we know the relation  in a hyperbola  Hence,The Equation of the hyperbola is ;

6.  Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas.

      49y^2 - 16x^2 = 784

Given a Hyperbola equation, Can also be written as   Comparing this equation with the standard equation of the hyperbola: We get,  and  Now, As we know the relation in a hyperbola, Therefore, Coordinates of the foci: The Coordinates of vertices: The Eccentricity: The Length of the latus rectum :

 5.  Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas.

     5y^2 - 9x^2 = 36

Given a Hyperbola equation, Can also be written as  Comparing this equation with the standard equation of the hyperbola: We get,    and  Now, As we know the relation in a hyperbola, Here as we can see from the equation that the axis of the hyperbola is Y-axis. So,  Coordinates of the foci: The Coordinates of vertices: The Eccentricity: The Length of the latus rectum :

 4.  Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas.

     16x^2 - 9y^2 = 576

Given a Hyperbola equation, Can also be written as  Comparing this equation with the standard equation of the hyperbola: We get,  and  Now, As we know the relation in a hyperbola, Therefore, Coordinates of the foci: The Coordinates of vertices: The Eccentricity: The Length of the latus rectum :

3.  Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas.

      9 y^2 - 4 x^2 =36

Given a Hyperbola equation, Can also be written as  Comparing this equation with the standard equation of the hyperbola: We get,  and  Now, As we know the relation in a hyperbola, Hence,  Coordinates of the foci: The Coordinates of vertices: The Eccentricity: The Length of the latus rectum :

2.  Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas.

      \frac{y^2}{9} - \frac{x^2}{27} = 1

Given a Hyperbola equation, Can also be written as  Comparing this equation with the standard equation of the hyperbola: We get,  and  Now, As we know the relation in a hyperbola, Here as we can see from the equation that the axis of the hyperbola is Y-axis. So,  Coordinates of the foci: The Coordinates of vertices: The Eccentricity: The Length of the latus rectum :

1.  Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas.

      \frac{x^2}{16} - \frac{y^2}{9} = 1

Given a Hyperbola equation, Can also be written as  Comparing this equation with the standard equation of the hyperbola: We get,  and  Now, As we know the relation in a hyperbola, Here as we can see from the equation that the axis of the hyperbola is X -axis. So,  Coordinates of the foci: The Coordinates of vertices: The Eccentricity: The Length of the latus rectum :
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