Question 1: The equation of the directrix of a Hyperbola is Its focus is and eccentricity . Find the equation of the hyperbola.

Answer:

Let be the focus and be a point on the Hyperbola.

Draw perpendicular from on the directrics. Then, by definition

This is the required equation of the hyperbola.

Question 2: Find the equation of the hyperbola whose

(i) focus is directrix is and eccentricity

(ii) focus is directrix is and eccentricity

(iii) focus is directrix is and eccentricity

(iv) focus is directrix is and eccentricity

(v) focus is directrix is and eccentricity

(vi) focus is directrix is and eccentricity

Answer:

(i) Given focus is directrix is and eccentricity

Let be the focus and be a point on the Hyperbola.

Draw perpendicular from on the directrics. Then, by definition

This is the required equation of the hyperbola.

(ii) Given focus is directrix is and eccentricity

Let be the focus and be a point on the Hyperbola.

Draw perpendicular from on the directrics. Then, by definition

This is the required equation of the hyperbola.

(iii) Given focus is directrix is and eccentricity

Let be the focus and be a point on the Hyperbola.

Draw perpendicular from on the directrics. Then, by definition

This is the required equation of the hyperbola.

(iv) Given focus is directrix is and eccentricity

Let be the focus and be a point on the Hyperbola.

Draw perpendicular from on the directrics. Then, by definition

This is the required equation of the hyperbola.

(v) Given focus is directrix is and eccentricity

Let be the focus and be a point on the Hyperbola.

Draw perpendicular from on the directrics. Then, by definition

This is the required equation of the hyperbola.

(vi) Given focus is directrix is and eccentricity

Let be the focus and be a point on the Hyperbola.

Draw perpendicular from on the directrics. Then, by definition

This is the required equation of the hyperbola.

Question 3: Find the eccentricity, coordinates of the foci, equations of directrices and length of the latus-rectum of the hyperbola

(i)

(ii)

(iii)

(iv)

(v)

Answer:

(i) Given

(ii) Given

(iii) Given

(iv) Given

(v) Given

Question 4: Find the axes, eccentricity, latus-rectum and the coordinates of the foci of the hyperbola

Answer:

Given