**Question 1.Find the(i) lengths of the principal axes(ii) co-ordinates of the foci(iii) equations of directrices(iv) length of the latus rectum(v) distance between foci(vi) distance between directrices of the ellipse:**

a = 5 and b = 3

Since a > b,

X-axis is the major axis and Y-axis is the minor axis.

(i) Length of major axis = 2a = 2(5) = 10

Length of minor axis = 2b = 2(3) = 6

Lengths of the principal axes are 10 and 6.

(i) Length of major axis = 2a = 2(2) = 4

Length of minor axis = 2b = 2√3

Lengths of the principal axes are 4 and 2√3.

a = √3 and b = 1

Since a > b,

X-axis is the major axis and Y-axis is the minor axis.

(i) Length of major axis = 2a = 2√3

Length of minor axis = 2b = 2(1) = 2

Lengths of the principal axes are 2√3 and 2.

**Question 2.**

Distance between foci = 2ae

Given, distance between foci = 8

2ae = 8

2(5)e = 8

(vi) Given, the length of the latus rectum is 6, and co-ordinates of foci are (±2, 0).

The foci of the ellipse are on the X-axis.

The ellipse passes through the points (-3, 1) and (2, -2).

Substituting x = -3 and y = 1 in equation of ellipse, we get

Equations (i) and (ii) become

9A + B = 1 …..(iii)

4A + 4B = 1 …..(iv)

Multiplying (iii) by 4, we get

36A + 4B = 4 …..(v)

Subtracting (iv) from (v), we get

32A = 3

The ellipse passes through (-√5, 2).

Substituting x = -√5 and y = 2 in equation of ellipse, we get

**Question 3.Find the eccentricity of an ellipse, if the length of its latus rectum is one-third of its minor axis.Solution:**

**Question 4.Find the eccentricity of an ellipse, if the distance between its directrices is three times the distance between its foci.Solution:**

**Question 5.**

**Question 6.**

**Question 7.**

**Question 8.**

**Question 9.**

**Question 10.**

**Question 11.Solution:**

Slope of the given line x + y + 1 = 0 is -1.

Since the given line is parallel to the required tangents,

the slope of the required tangents is m = -1.

Equations of tangents to the ellipse

**Question 12.**

Alternate method:

The locus of the point of intersection of perpendicular tangents is the director circle of an ellipse.

Question 13.

**Question 14.Show that the locus of the point of intersection of tangents at two points on an ellipse, whose eccentric angles differ by a constant, is an ellipse.Solution:**

**Question 15.**

**Question 16.**

**Question 17.**

**Question 18.**