SOLUTION: State the position of each line with respect to the ellipse 9x^2+25y^2 =225 a.)y=3x-2 b.)y=-x+6 c.)3x-4y+1=0 please help. and thank you in advance

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Question 1016128: State the position of each line with respect to the ellipse 9x^2+25y^2 =225
a.)y=3x-2
b.)y=-x+6
c.)3x-4y+1=0
please help. and thank you in advance
Found 2 solutions by Fombitz, ikleyn:
Answer by Fombitz(32388)   (Show Source): You can put this solution on YOUR website!
Graph it.
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Answer by ikleyn(52802)   (Show Source): You can put this solution on YOUR website!
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state the position of each lie with respect to the ellipse x^2/9 + y^2/36 = 1
a.)y=2x+2
b.)y=-2x-9
c.)2x+5y+3=0

its differenet from the first one that i already asked. im not really good at math please help. and thank you again in advance
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The way to solve this problem is as follows:


1. Express y via x from the given linear equation.

   (In your case it is just done in a.) and b.) ).


2. Substitute this expression into the equation for ellipse.

   You will get a quadratic equation.


3. If this quadratic equation has two real roots, then the straight line intersects the ellipse in two points.

   If this quadratic equation has one real root, then the straight line is tangent to the ellipse. 

   If this quadratic equation has no real roots, then the straight line has no common points with the ellipse. 
   I.e. the straight line is outside the ellipse.


4. Therefore, when you got the quadratic equation, calculate and check its discriminant.

   If the discriminant is positive, then the quadratic equation has two real roots. 
   Hence, the straight line intersects the ellipse in two points.

   If the discriminant is zero, then the quadratic equation has one real root. 
   Hence, the straight line is tangent to the ellipse. 

   If the discriminant is negative, then the quadratic equation has no real roots. 
   Hence, the straight line has no common points with the ellipse. I.e. the straight line is outside the ellipse.


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