Sample Questions on Hyperbola Formula

Question 1: Find the eccentricity of hyperbola having the equation x²/36 – y²/49 = 1.

Answer: 

Given equation is x²/36 – y²/49 = 1.

Comparing with equation of the hyperbola x²/a² – y²/b² = 1

Where a² = 36 and b² = 49

As we know formula for eccentricity,

e = √( 1+ b²/a² )

= √( 1 + 49/ 36 )

= √( 36+49/36 )

= √(85/36)

= √85/√36

= 9.21/6

= 1.53

Therefore we got eccentricity for x²/36 – y²/49 = 1 is 1.53.

Question 2: Find the eccentricity of hyperbola having the equation x²/27 – y²/25 = 1.

Answer:

Given equation is  x²/27 – y²/25 = 1

Comparing with equation of the hyperbola x²/a² – y²/b² = 1

Where a² = 27 and b² = 25

As we know formula for eccentricity,

e = √( 1+ b²/a² )

= √( 1 + 25/ 27 )

= √( 27+25/27 )

= √(52/27)

= √52/√27

= 7.21/5.19

= 1.38

Therefore we got eccentricity for x²/36 – y²/49 = 1 is 1.38.

Question 3: A hyperbola has an eccentricity of 1.3 and the value of a is 20. Find the hyperbola’s equation.

Answer:

Given eccentricity is 1.38 and value for a is  20

As we know the formula for eccentricity,

e = √( 1+ b²/a² )

then

1.3 =  √(1+b²/20²)

13/10 =  √( 400 + b²/ 400 )

(13/10)² = ( 400 +b²/400 )

169/100 = ( 400 +b²/400 )

b²= 276

Comparing with equation of the hyperbola x²/a² – y²/b² = 1

then,

x²/400 – y²/276 = 1

Question 4: State what is a Hyperbola?

Answer:

The locus of a point moving in a plane where the ratio of its distance from a fixed point to that from a fixed-line is a constant greater than one is called a hyperbola.

Question 5: What is the directrix of hyperbola and its formula?

Answer: 

A hyperbola’s directrix is a straight line that is utilized to generate a curve. It is also known as the line away from which the hyperbola curves. The symmetry axis is perpendicular to this line. The directrix equation is:

x = ±a² /√(a² + b²)

Question 6: What is the formula for conjugate hyperbola?

Answer:

Two hyperbolas whose transverse and conjugate axes are the conjugate and transverse axes of the other are referred to as conjugate hyperbolas of each other.

e = √( 1+ a²/b² )

Hyperbola Formula: The set of all points in a plane is called a hyperbola. The distance between these two fixed points in the plane will remain constant. The distance to the distant location minus the distance to the nearest point is the difference. The foci will be the two fixed points, and the center of the hyperbola will be the mid-point of the line segment connecting the foci. Hyperbola is a fascinating topic in geometrical mathematics.

This article explores the hyperbola formulas, along with their equations, and solved examples on it.