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3 years ago

The equation below describes a parabola. If a is negative, which way does the parabola open? x = ay2

Mathematics

2 answers:

Anastasy [175]3 years ago

8 0

This type of parabola opens either to the left or to the right. The negative makes it open to the left.

Vanyuwa [196]3 years ago

3 0

Answer:

The given parabola opens to the left.

Step-by-step explanation:

We are given,

The equation of the parabola is $x=ay^2$

If in the parabola,

1. If 'a' is positive, then the parabola opens upwards or right.

2. If 'a' is negative, then the parabola opens downwards or left.

Moreover,

3. If x is squared, then the parabola opens upwards or downwards.

4. If y is squared, then the parabola opens left or right.

Now, as we are given the parabola having $y^2$ and 'a' is negative.

Thus, from the points 4 and 2, we get,

The given parabola opens to the left.

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LORAN is a long range hyperbolic navigation system. Suppose two LORAN transmitters are located at the coordinates (−60,0) and (6

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LORAN follows an hyperbolic path.

The equation of the hyperbola is: $\mathbf{\frac{x^2}{2500} + \frac{y^2}{1100} = 1}$

The coordinates are given as:

$\mathbf{(x,y) = (-60,0)\ (60,0)}$

The center of the hyperbola is

$\mathbf{(h,k) = (0,0)}$

The distance from the center to the focal points is given as:

$\mathbf{c = 60}$

Square both sides

$\mathbf{c^2 = 3600}$

The distance from the receiver to the transmitters is given as:

$\mathbf{2a = 100}$

Divide both sides by 2

$\mathbf{a = 50}$

Square both sides

$\mathbf{a^2 = 2500}$

We have:

$\mathbf{b^2 = c^2 - a^2}$

This gives

$\mathbf{b^2 = 3600 - 2500}$

$\mathbf{b^2 = 1100}$

The equation of an hyperbola is:

$\mathbf{\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1}$

So, we have:

$\mathbf{\frac{(x - 0)^2}{2500} + \frac{(y - 0)^2}{1100} = 1}$

$\mathbf{\frac{x^2}{2500} + \frac{y^2}{1100} = 1}$

Hence, the equation of the hyperbola is: $\mathbf{\frac{x^2}{2500} + \frac{y^2}{1100} = 1}$