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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 :

20. Find the equation for the ellipse that satisfies the given conditions:

     Major axis on the x-axis and passes through the points (4,3) and (6,2).

Given, in an ellipse Major axis on the x-axis and passes through the points (4,3) and (6,2). Since The major axis of this ellipse is on the  X-axis, the equation of the ellipse will be of the form:   Where  and are the length of the semimajor axis and semiminor axis respectively. Now since the ellipse passes through the point,(4,3) since the ellipse also passes through the point (6,2). On...

18. Find the equation for the ellipse that satisfies the given conditions:

       b = 3, c = 4, centre at the origin; foci on the x axis.

Given,In an ellipse,    b = 3, c = 4, centre at the origin; foci on the x axis. Here  foci of the ellipse are in X-axis so the major axis of this ellipse will be X-axis. Therefore, the equation of the ellipse will be of the form:   Where  and are the length of the semimajor axis and semiminor axis respectively. Also Given,   and  Now, As we know the relation, Hence, The Equation of the...

17.  Find the equation for the ellipse that satisfies the given conditions:

       Foci (± 3, 0), a = 4

Given, In an ellipse,  V Foci (± 3, 0), a = 4 Here foci of the ellipse are in X-axis so the major axis of this ellipse will be X-axis. Therefore, the equation of the ellipse will be of the form:   Where  and are the length of the semimajor axis and semiminor axis respectively. So on comparing standard parameters( vertices and foci) with the given one, we get   and  Now, As we know the...

 16.  Find the equation for the ellipse that satisfies the given conditions:

     Length of minor axis 16, foci (0, ± 6).

Given, In an ellipse,   Length of minor axis 16, foci (0, ± 6). Here, the focus of the ellipse is on the  Y-axis so the major axis of this ellipse will be Y-axis. Therefore, the equation of the ellipse will be of the form:   Where  and are the length of the semimajor axis and semiminor axis respectively. So on comparing standard parameters( length of semi-minor axis and foci) with the given...

15. Find the equation for the ellipse that satisfies the given conditions:

       Length of major axis 26, foci (± 5, 0)

Given, In an ellipse,  Length of major axis 26, foci (± 5, 0) Here, the focus of the ellipse is in X-axis so the major axis of this ellipse will be X-axis. Therefore, the equation of the ellipse will be of the form:   Where  and are the length of the semimajor axis and semiminor axis respectively. So on comparing standard parameters( Length of semimajor axis and foci) with the given one, we...

14. Find the equation for the ellipse that satisfies the given conditions:

      Ends of major axis (0, ± \sqrt{5} ), ends of minor axis (± 1, 0)

Given, In an ellipse,   Ends of the major axis (0, ± ), ends of minor axis (± 1, 0) Here, the major axis of this ellipse will be Y-axis. Therefore, the equation of the ellipse will be of the form:   Where  and are the length of the semimajor axis and semiminor axis respectively. So on comparing standard parameters( ends of the major and minor axis ) with the given one, we get   and  Hence, The...

13. Find the equation for the ellipse that satisfies the given conditions: 

       Ends of major axis (± 3, 0), ends of minor axis (0, ± 2)

Given, In an ellipse,  Ends of the major axis (± 3, 0), ends of minor axis (0, ± 2) Here, the major axis of this ellipse will be X-axis. Therefore, the equation of the ellipse will be of the form:   Where  and are the length of the semimajor axis and semiminor axis respectively. So on comparing standard parameters( ends of the major and minor axis ) with the given one, we get   and  Hence, The...

12.  Find the equation for the ellipse that satisfies the given conditions:

      Vertices (± 6, 0), foci (± 4, 0)

Given, In an ellipse,   Vertices (± 6, 0), foci (± 4, 0) Here Vertices and focus of the ellipse are in X-axis so the major axis of this ellipse will be X-axis. Therefore, the equation of the ellipse will be of the form:   Where  and are the length of the semimajor axis and semiminor axis respectively. So on comparing standard parameters( vertices and foci) with the given one, we get   and  Now,...

11. Find the equation for the ellipse that satisfies the given conditions:

       Vertices (0, ± 13), foci (0, ± 5)

Given, In an ellipse,   Vertices (0, ± 13), foci (0, ± 5) Here Vertices and focus of the ellipse are in Y-axis so the major axis of this ellipse will be Y-axis. Therefore, the equation of the ellipse will be of the form:   Where  and are the length of the semimajor axis and semiminor axis respectively. So on comparing standard parameters( vertices and foci) with the given one, we...

10. Find the equation for the ellipse that satisfies the given conditions:

       Vertices (± 5, 0), foci (± 4, 0)

Given, In an ellipse,  Vertices (± 5, 0), foci (± 4, 0) Here Vertices and focus of the ellipse are in X-axis so the major axis of this ellipse will be X-axis. Therefore, the equation of the ellipse will be of the form:   Where  and are the length of the semimajor axis and semiminor axis respectively. So on comparing standard parameters( vertices and foci) with the given one, we get   and  Now,...

9. Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.

       4x^2 + 9y^2 =36

Given The equation of the ellipse As we can see from the equation, the major axis is along X-axis and the minor axis is along Y-axis. On comparing the given equation with the standard equation of an ellipse, which is  We get   and . So, Hence, Coordinates of the foci:   The vertices: The length of the major axis: The length of minor axis: The eccentricity : The length of the latus rectum:

8. Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.

       16x^2 + y^2 = 16

Given The equation of the ellipse As we can see from the equation, the major axis is along Y-axis and the minor axis is along X-axis. On comparing the given equation with the standard equation of such  ellipse, which is  We get   and . So, Hence, Coordinates of the foci:   The vertices: The length of the major axis: The length of minor axis: The eccentricity : The length of the latus rectum:

7. Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.

       36x^2 + 4y^2 =144

Given The equation of the ellipse As we can see from the equation, the major axis is along Y-axis and the minor axis is along X-axis. On comparing the given equation with the standard equation of such  ellipse, which is  We get   and . So, Hence, Coordinates of the foci:   The vertices: The length of the major axis: The length of minor axis: The eccentricity : The length of the...

6. Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.

       \frac{x^2}{100} + \frac{y^2}{400} =1

Given The equation of the ellipse As we can see from the equation, the major axis is along Y-axis and the minor axis is along X-axis. On comparing the given equation with the standard equation of such  ellipse, which is  We get   and . So, Hence, Coordinates of the foci:   The vertices: The length of the major axis: The length of minor axis: The eccentricity : The length of the latus rectum:

5.  Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.

       \frac{x^2}{49} + \frac{y^2}{36} = 1

Given The equation of ellipse As we can see from the equation, the major axis is along X-axis and the minor axis is along Y-axis. On comparing the given equation with standard equation of ellipse, which is  We get   and . So, Hence, Coordinates of the foci:   The vertices: The length of major axis: The length of minor axis: The eccentricity : The length of the latus rectum:

4. Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.

       \frac{x^2}{25} + \frac{y^2}{100} = 1

Given The equation of the ellipse As we can see from the equation, the major axis is along Y-axis and the minor axis is along X-axis. On comparing the given equation with the standard equation of such  ellipse, which is  We get   and . So, Hence, Coordinates of the foci:   The vertices: The length of the major axis: The length of minor axis: The eccentricity : The length of the latus rectum:

3.  Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.

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

Given The equation of the ellipse As we can see from the equation, the major axis is along X-axis and the minor axis is along Y-axis. On comparing the given equation with the standard equation of an ellipse, which is  We get   and . So, Hence, Coordinates of the foci:   The vertices: The length of the major axis: The length of minor axis: The eccentricity : The length of the latus rectum:

2. Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.

       \frac{x^2}{4} + \frac{y^2}{25} =1

Given The equation of the ellipse As we can see from the equation, the major axis is along Y-axis and the minor axis is along X-axis. On comparing the given equation with the standard equation of such  ellipse, which is  We get   and . So, Hence, Coordinates of the foci:   The vertices: The length of the major axis: The length of minor axis: The eccentricity : The length of the latus rectum:
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