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Question

1. Which is the only conic section that has asymptotes?
2. Is the value of B2 – 4AC for the hyperbola greater than, less than, or equal to zero?
3. What method was used to factor the equations that had factorable left-hand sides? 

Answer 1

(a)

Asymptotes : A line whose distance to a given curve tends to zero as x tends to infinity.

 

In the conic section, Hyperbola is the only one which has asymptotes.

 

Asymptote of Hyperbola is .

 

Answer 2

(b)

General Equation of conic : Ax² + Bxy + Cy² + Dx + Ey + F = 0

Conditions :

B² – 4AC < 0 then Equation represents Circle or Ellipse

B² – 4AC > 0 then Equation represents Hyperbola

B² – 4AC = 0 then Equation represents Parabola.

Therefore for Hyperbola, B² – 4AC > 0 .

Answer 3

(c)

We can make use of the method "Completing the squares" to  factor the equation.

" Completing the squares " is a method in which we represent the given conic function in its standard form.

Let us consider an example

Parabola equation is y = x² - 2x - 2

To change the expression (x² - 2x) into a perfect square trinomial add (half the x coefficient)² to each side of the expression.

y = (x² - 2x - 2 + 1 - 1)

y = (x² - 2x + 1 - 2 - 1)

y = (x² - 2x + 1 - 3)

y  = (x - 1)² - 3

The vertex form of the equation of parabola is y  = 1(x + 1)² - 6.