Everything Maths Grade 10

Worked example 23: Determining the equation of a hyperbola

QUESTION

Use the sketch below to determine the values ofaandqfor the hyperbola of the formy=ax+q.

(^0) (1; 0)
(1; 2)
x
y

SOLUTION

Step 1: Examine the sketch

The two curves of the hyperbola lie in the second and fourth quadrant, thereforea < 0. We also see that the
graph has been shifted vertically upwards, thereforeq > 0.

Step 2: Substitute the given points into the equation and solve
Substitute the point(1; 2):

y=

a

x

+q

2 =

a

 1

+q

)2 =a+q

Substitute the point(1; 0):

y=

a

x

+q

0 =

a

1

+q

)a=q

Step 3: Solve the equations simultaneously using substitution

2 =a+q

=q+q

= 2q

)q= 1

)a=q

= 1

Step 4: Write the final answer

a= 1 andq= 1, therefore the equation of the hyperbola isy=

1

x

+ 1.

Chapter 6. Functions 209