*Note: Also plotted the ellipse equations. Students should have a feel of what the equation looks like. *

Question 1: Find the equation of the ellipse whose focus is the directrix and eccentricity equal to

Answer:

Let be any point on the ellipse whose focus is and eccentricity

Let be perpendicular from on directrix. Then,

This is the required equation of the ellipse.

Question 2: Find the equation of the ellipse in the following cases:

Answer:

(i) Let be a point on ellipse. Then, by definition

This is the required equation of the ellipse.

(ii) Let be a point on ellipse. Then, by definition

This is the required equation of the ellipse.

(iii) Let be a point on ellipse. Then, by definition

This is the required equation of the ellipse.

(iv) Let be a point on ellipse. Then, by definition

This is the required equation of the ellipse.

Question 3: Find the eccentricity, coordinates of foci, length of the latus-rectum of the following ellipse:

(i)

(ii)

(iii)

(iv)

(v)

Answer:

(i)

(ii)

(iii)

(iv)

(v)

Question 4: Find the equation to the ellipse (referred to its axes as the axes of and respectively which passes through the point and has eccentricity

Answer:

Substituting this in equation i) we get

Therefore the equation of ellipse is

This is the required equation of ellipse.

Question 5: Find the equation of the ellipse in the following cases:

Answer:

Coordinates of the foci

Squaring both sides, we get:

This is the required equation of the ellipse.

Squaring both sides, we get:

This is the required equation of the ellipse.

Squaring both sides, we get:

This is the required equation of the ellipse.

Squaring both sides, we get:

This is the required equation of the ellipse.

Ellipse passes through

Solving i) and ii) we get

Substituting the values, we get:

This is the required equation of the ellipse.

Let the equation of the required ellipse be

The coordinates of its vertices and foci are and respectively.

Substituting the values of and in ( i) we get,

This is the required equation of the ellipse.

Let the equation of the required ellipse be

The coordinates of its vertices and foci are and respectively.

Substituting the values of and in ( i) we get,

This is the required equation of the ellipse.

Let the equation of the required ellipse be

The coordinates of its vertices and foci are and respectively.

Substituting the values of and in ( i) we get,

This is the required equation of the ellipse.

Let the equation of the required ellipse be

End of major axis

End of minor axis

The coordinates of the end points of the major and minor axes are and respectively.

This is the required equation of the ellipse.

Let the equation of the required ellipse be

End of major axis

End of minor axis

The coordinates of the end points of the major and minor axes are and respectively.

This is the required equation of the ellipse.

Let the equation of the required ellipse be

Squaring both sides we get:

This is the required equation of the ellipse.

Let the equation of the required ellipse be

Squaring both sides we get:

This is the required equation of the ellipse.

Squaring both sides we get:

This is the required equation of the ellipse.

Question 6: Find the equation of the eclipse whose foci are (4,0) and ( -4, 0) , eccentricity

Answer:

Let the equation of the required ellipse be

The coordinates of the foci are

Also

We have

Now,

Substituting the values of and in equation (i), we get:

This is the required equation of the ellipse.

Question 7: Find the equation of the-ellipse in the standard form whose minor axis is equal to the distance between foci and whose latus-rectum is 10.

Answer:

Given minor axis is equal to the distance between foci and whose latus-rectum is 10.

Substituting the values of a and b in the equation of an ellipse, we get:

This is the required equation of the ellipse.

Question 8: Find the equation of the ellipse whose center is (- 2,3) and whose semi-axis are 3 and 2 when major axis is (i) parallel to x-axis (ii) parallel to y-axis

Answer:

(i) Let and be the major and minor axes of the ellipse. Then, the equation is

Given center is

We have semi major axis

Similarly, semi major axis

Putting and in equation (i) we get

This is the required equation of the ellipse.

(ii) Let and be the major and minor axes of the ellipse. Then, the equation is

We have semi major axis

Similarly, semi major axis

Given center is

Putting and in equation (i) we get

This is the required equation of the ellipse.

Question 9: Find the eccentricity of an ellipse whose latus-rectum is:

(i) half of its minor axis (ii) half of its major axis.

Answer:

Let and be the major and minor axes of the ellipse.

(i) When latus-rectum is half of minor axis

Now,

(ii) When latus-rectum is half of minor axis

Now,

Question 10: Find the center, the lengths of the axes, eccentricity, foci of the following ellipse:

Answer:

Here, and

The coordinates of the center of the ellipse are

Clearly, so, the given equation represents an ellipse whose major and minor axes are along the X and Y axes respectively.

Eccentricity: The eccentricity is given by

Here, and

The coordinates of the center of the ellipse are

Clearly, so, the given equation represents an ellipse whose major and minor axes are along the X and Y axes respectively.

Eccentricity: The eccentricity is given by

Here, and

The coordinates of the center of the ellipse are

Clearly, so, the given equation represents an ellipse whose major and minor axes are along the Y and X axes respectively.

Eccentricity: The eccentricity is given by

Here, and

The coordinates of the center of the ellipse are

Clearly, so, the given equation represents an ellipse whose major and minor axes are along the X and Y axes respectively.

Eccentricity: The eccentricity is given by

Here, and

The coordinates of the center of the ellipse are

Eccentricity: The eccentricity is given by

Here, and

The coordinates of the center of the ellipse are

Eccentricity: The eccentricity is given by

Question 11: Find the equation of an ellipse whose foci are at and which passes through

Answer:

Let the equation of the required ellipse be

The coordinates of its foci are

The required ellipse passes through

Substituting this in equation ii) we get

(because then would be negative which is not possible).

Therefore

Therefore the required equation of the ellipse is

This is the required equation of the ellipse.

Question 12: Find the equation of an ellipse whose eccentricity is latus-rectum is and the center is at the origin.

Answer:

Let the equation of the required ellipse be

This is the required equation of the ellipse.

Question 13: Find the equation of an ellipse with its foci on y-axis, eccentricity center at the origin and passing through

Answer:

Let the equation of the ellipse be

Substituting in equation i) we get

This is the equation of the required ellipse.

Question 14: Find the equation of an ellipse whose axes lie along coordinate axes and which passes through and

Answer:

Let the equation of the required ellipse be

The required ellipse passes through and

Solving equation ii) and iii) we get

Substituting in equation i) we get

This is the equation of the required ellipse.

Question 15: Find the equation of an ellipse whose axes lie along the coordinate axes, which passes through the point and has eccentricity equal to

Answer:

Let the equation of the required ellipse be

The required ellipse passes through

Substituting ii) into iii) we get

Substituting in ii) we get

Substituting in i) we get

This is the equation of the required ellipse.

Question 16: Find the equation of an ellipse, the distance between the foci is 8 units and the distance between the directrices is 18 units.

Answer:

Let the equation of the required ellipse be

The distance between foci is units

The distance between the directrics is units

From ii) and iii) we get

Substituting and in equation i) we get

This is the equation of the required ellipse.

Question 17: Find the equation of an ellipse whose vertices are and eccentricity

Answer:

Let the equation of the required ellipse be

The coordinates of the vertices are

Substituting and in equation i) we get

This is the equation of the required ellipse.

Question 18: A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with x-axis.

Answer:

Let be the rod making an angle with and let be the point on it such that

Question 19: Find the equation of the set of all points whose distances from are of their distances from the line

Answer: