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{x^2}/{3^2} + {y^2}/{7^2} = 1if x= 4+5cost what is y?

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2 Answers

Dear lizzy

x2/32 + y2/72 = 1 is an equation for an ellipse and it's major axis is y axis and it's minor axis is x axis.

If you draw the figure for this ellipse, the x value varies from -3 to 3 and y value varies from -7 to 7.

If x value is bigger than 3 then there is no y value for this equation in real number set. For example, if you think x=6 then y2/72 = -3, and in this situation y square is negative value and y is not real number.

 

But they gave you a equation x=4+5cost. This x value varies from -1 to 9 because cost varies from -1 to 1.

So, if -1 ≤ x ≤ 3 ( this is the same as -1 ≤ cost ≤ -1/5 and again this is the same as  1.77 ≤ t ≤ 4.51 [rad] ) there are two solusions for y, such as y = ±√c form. ( you just plug in x value which decided by x = 4 +5cost to y = ±7√(1-x2/32) )

But if x > 3 ( this is the same as cost > -1/5 and again this is the same as 0 ≤ t < 1.77 or 4.51 < t ≤ 2pi ) there is no solution for y in the real number.

 

If this is a high level math problem such as olypiad, then this problem can be accepted but for general school level, this problem gave you the wrong x values to you.

If you need more explanation please feel free.

 

This question has a lot of boring algebra that's easy to get caught up in, so I'll let you work that part out.  Basically, what they're wanting you to do is to break up this ellipse into two parametric equations.  They give you one, x = 4 + 5cost.  You plug that in for x and solve for y, and you will end up with an equation for y in terms of t.  So what you're looking for is going to be something like y = 8 - 2sint.  (That's not your answer, just an example of what your answer may look like)

So, your first step is to do the substitution, like so:

(4 + 5cost)2 + y2 = 1
      32               72

From there, solve for y.