Hyperbolas V

Question

What is the value of c? 
5x² - 3y² - 20x + 6y + 2 = 0

Select one:

a. 2

b. 4

c. √4

d. √8

How is the transverse axis oriented for the following equation?
(y - 5)² + (x - 3)²
    7             6

Select one:

a. Horizontal

b. Vertical

c. Perpendicular

d. Parallel

 

Answer 1

1)

The hyperbola equation is 5x² - 20x - 3y² + 6y + 2 = 0.
Write the equation : 5x² - 20x - 3y² + 6y + 2 = 0. in standard form of hyperbola : .

5x² - 20x - 3y² + 6y + 2 = 0.

5(x² - 4x) - 3(y² - 2y) + 2 = 0

To change the expression into a perfect square  add (half the x coefficient)² and (half the y - coefficient)² to each side of the expression.

Here x coefficient = - 4, so, (half the x coefficient)² = (- 4/2)²= 4.

Here y coefficient = - 2, so, (half the x coefficient)² = (- 2/2)²= 1.

5(x² - 4x + 4) - 20 - 3(y² - 2y + 1) + 3 + 2= 0

5(x - 2)² - 3(y - 1)² = 15

(x - 2)² /3- (y - 3)²/5 = 1

Compare the above equation with standard form of hyperbola equation.

c² = a²+ b².

c² = 3 +  5

⇒ c = √8 .

Option d is correct.

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2)

The standard form of the equation of a hyperbola with center at the (h,k) (where a and b are not equals to 0) is (x-h)²/a² - (y-k)²/b² = 1 (Transverse axis is horizontal) or (y-k)²/a² -( x-h)²/b² = 1 (Transverse axis is vertical).

Compare the given equation (y - 5)² + (x - 3)²
                                          7             6

The given equation is in the form  (y-k)²/a² -( x-h)²/b² = 1

So Transverse axis is vertical.

Option (b) is correct.