4x^2 = 7y How do they get the following for the focus and directix?

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In summary, the conversation discusses finding the focus and directrix of a parabola with the given values of focus and directrix. The formula for finding the focus and directrix is given, but it does not work for the given equation. The expert advises correctly applying the formula by dividing both sides by the coefficient of the x^2 term.
  • #1
Jurrasic
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Please don't leave out very many steps, please do give the formulas if possible please thank you:
How did they get these values?
Focus = (0,7/16)
directrix = -7/16

Tried to use p=1/4(a) which DID NOT work , that yields p = 1
THEN the focus would be (0,1) which it is not. Teacher said to use that formula but maybe that was for who knows what , because it's not working.
 
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  • #2
[itex]4x^2=7y \rightarrow 4x^2+0x+0=7y \rightarrow (4/7)x^2+0x+0=y[/itex]
thus, the focus is
[itex]
(-\frac{0}{2\cdot(\frac{4}{7})}, -\frac{0^2}{4\cdot(\frac{4}{7})}+0+\frac{1}{4\frac{4}{7}} ) = ( 0, \frac{7}{16} )
[/itex]
the same thing goes with the directrix.

hope it helped.
 
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  • #3
^No offense, but I don't know how that post could have helped. It was hard for me to read.

Jurrasic said:
Tried to use p=1/4(a) which DID NOT work , that yields p = 1
THEN the focus would be (0,1) which it is not. Teacher said to use that formula but maybe that was for who knows what , because it's not working.
You are not applying the formula correctly. One of the equations of the parabola is
x2 = 4py,
but your equation is
4x2 = 7y.
You have a coefficient for the x2 term. So the first thing you must do is to divide both sides by 4.
[itex]x^2 = \frac{7}{4}y[/itex]
Now you can figure out the focus and directrix.
 

FAQ: 4x^2 = 7y How do they get the following for the focus and directix?

What is the focus of the parabola represented by the equation 4x^2 = 7y?

The focus of a parabola is a point that is equidistant from the directrix and the vertex. To find the focus of the parabola represented by the equation 4x^2 = 7y, we first need to rewrite the equation in the standard form (y - k) = a(x - h)^2. In this form, the coordinates of the focus are (h, k + 1/4a). Therefore, the focus of the parabola 4x^2 = 7y is (0, 7/16).

What is the directrix of the parabola represented by the equation 4x^2 = 7y?

The directrix of a parabola is a line that is perpendicular to the axis of symmetry and is located at a distance a from the vertex. To find the directrix of the parabola represented by the equation 4x^2 = 7y, we need to solve for y and then find the value of y when x = 0. This will give us the equation of the directrix, which is a horizontal line passing through the point (0, 7/4). Therefore, the directrix of the parabola 4x^2 = 7y is y = 7/4.

How do you determine the orientation of the parabola represented by the equation 4x^2 = 7y?

The orientation of a parabola is determined by the coefficient of the x^2 term in the equation. If this coefficient is positive, the parabola opens upwards and if it is negative, the parabola opens downwards. Therefore, in the equation 4x^2 = 7y, the coefficient of x^2 is positive, indicating that the parabola opens upwards.

Can you graph the parabola represented by the equation 4x^2 = 7y?

Yes, we can graph the parabola represented by the equation 4x^2 = 7y by first finding the vertex, focus, and directrix as mentioned above. Then, we can plot these points on a coordinate plane and draw a smooth curve passing through the vertex and focus, and perpendicular to the directrix. Alternatively, we can also use a graphing calculator to plot the parabola.

How can the equation 4x^2 = 7y be used in real life situations?

The equation 4x^2 = 7y represents a parabola, which is a common shape found in nature and used in various fields such as physics, engineering, and economics. For example, the equation can be used to model the trajectory of a projectile, the shape of a suspension bridge, or the profit function of a business. It can also be used in optics to describe the shape of a satellite dish or a parabolic mirror. In short, the equation has various real-life applications and is an important tool for solving many problems in different fields.

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