Show that (3 / 2, 3 / 2) is the only solution of {x + y = 3, 4xy = 9}. As well, show that the line and the curve hit each other at (3 / 2, 3 / 2).

x + y = 3

4xy = 9

y = 3 - x

4x(3 - x) = 9

4x² - 12x + 9 = 0

(2x - 3)² = 0

x = 3 / 2

y = 3 - (3 / 2)

y = 3 / 2

only solution as (3 / 2, 3 / 2)

tangent to 4xy = 9 at (3 / 2, 3 / 2) as x + y = 3

 

Show that (5 / 2, 5 / 2) is the only solution of {x + y = 5, 4xy = 25}. As well, show that the line and the curve hit each other at (5 / 2, 5 / 2).

x + y = 5

4xy = 25

y = 5 - x

4x(5 - x) = 25

4x² - 20x + 25 = 0

(2x - 5)² = 0

x = 5 / 2

y = 5 - (5 / 2)

y = 5 / 2

only solution as (5 / 2, 5 / 2)

tangent to 4xy = 25 at (5 / 2, 5 / 2) as x + y = 5

 

Show that (5 / 3, 5 / 3) is the only solution of {3x + 3y = 10, 9xy = 25}. As well, show that the line and the curve hit each other at (5 / 3, 5 / 3).

3x + 3y = 10

9xy = 25

3y = 10 - 3x

y = (10 - 3x) / 3

9x(10 - 3x) / 3 = 25

9x² - 30x + 25 = 0

(3x - 5)² = 0

x = 5 / 3

y = {10 - 3(5 / 3)} / 3

y = 5 / 3

only solution as (5 / 3, 5 / 3)

tangent to 9xy = 25 at (5 / 3, 5 / 3) as 3x + 3y = 10