In the previous article , we solved few solutions of Short Answer Type Questions of Ellipse Chapter of S.N.Dey mathematics, Class 11. In this chapter, we will solve few more.
4(i) Find the lengths of axes of the ellipse whose eccentricity is and the distance between focus and directrix is
Solution.
The distance between focus and directrix
By question,
Hence, the length of the major axis and the length of the minor axis
and are two foci of an ellipse whose eccentricity is Find the length of the major axis.
Solution.
Eccentricity
The distance between the two given foci is
Hence, the length of the major axis
5. The length of the latus rectum of an ellipse is unit and that of the major axis, which lies along the axis , is unit. Find its equation in the standard form . Determine the co-ordinates of the foci and the equations of its directrices.
Solution.
The length of the latus rectum of the ellipse is given by
Again, the length of the major axis (2a) is given by
Hence,the equation of the ellipse is
The eccentricity of the ellipse is given by
The co-ordinates of the foci of the ellipse is given by
The equation of the directrix is given by
6. Taking major and minor axes along and axes, find the equation of the ellipse whose
co-ordinates of foci are and the length of minor axis is
Solution.
Since the foci of the given ellipse lie on the -axis , the major axis of the ellipse lies on the axis.
Again, the centre of the ellipse is given by i.e.
So, the ellipse is of the form
So, the co-ordinates of the foci
The length of the minor axis is given by
Hence, using , the equation of the ellipse is
eccentricity and the length of latus rectum
Solution.
Length of latus rectum
From and we get,
So, the equation of the ellipse is
length of minor axis is and the distance between the foci is
Solution.
By the condition, length of minor axis is given by
The distance between the foci is
Hence, the equation of the ellipse is given by
co-ordinates of one vertex are and the co-ordinates of one end of minor axis are
Solution.
By question, the equation of the ellipse can be taken as
Co-ordinates of one vertex is and co-ordinates of one end of minor axis is
Hence, by we get,
co-ordinates of foci are and the eccentricity is
Solution.
By question, the co-ordinates of foci
Hence, the equation of the ellipse is
Find the equation of the ellipse whose
eccentricity is focus is and directrix is
Solution.
Let be any point on the ellipse . The co-ordinates of the focus : and the equation of the directrix is given by
Now, the length of the perpendicular from P on the straight line (1) is given by
,
Now, for the ellipse, we have
Hence, the equation of the ellipse is given by
eccentricity is focus is and directrix is
Solution.
Let be any point on the ellipse. The co-ordinates of the focus :
The equation of the directrix is
The length of the perpendicular from P on the straight line (1) is
,
Now, for the ellipse
Hence, the equation of the ellipse is
eccentricity is focus is directrix is
Solution.
Let be any point on the ellipse. The co-ordinates of focus : The equation of the directrix :
The perpendicular distance of the point P from the straight line (1) is
Now, for the ellipse, we know that
focus is directrix is and eccentricity is
Solution.
Let be any point on the ellipse. The co-ordinates of focus : The equation of the directrix :
The perpendicular distance of the point P from the straight line (1) is
Now, for the ellipse, we know that
Hence, the equation represents the required ellipse.