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Let E be the ellipse given by the equation X^2+7y^2=8
1) if m is any real number, find all tangent lines to E that pass throught the point (m,0)
2) the ellipse E has a tangent line with postive slope that passes throught the point (8,0) find the point of intersection of this line with the vertical line at x=13 need help please
 2 years ago
 2 years ago
Let E be the ellipse given by the equation X^2+7y^2=8 1) if m is any real number, find all tangent lines to E that pass throught the point (m,0) 2) the ellipse E has a tangent line with postive slope that passes throught the point (8,0) find the point of intersection of this line with the vertical line at x=13 need help please
 2 years ago
 2 years ago

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satellite73Best ResponseYou've already chosen the best response.2
i guess we need the derivative first
 2 years ago

satellite73Best ResponseYou've already chosen the best response.2
\[2x+14yy'=0\] \[y'=\frac{x}{7y}\]
 2 years ago

2bornot2bBest ResponseYou've already chosen the best response.1
This is a wonderful problem satellite
 2 years ago

2bornot2bBest ResponseYou've already chosen the best response.1
Take a look, the point (m,0) may not lie on the ellipse, are you getting the complexity?
 2 years ago

satellite73Best ResponseYou've already chosen the best response.2
oh then i probably messed up somewhere
 2 years ago

2bornot2bBest ResponseYou've already chosen the best response.1
I didn't mean you messed, I just wanted to let you know that the question is good
 2 years ago

sandman1Best ResponseYou've already chosen the best response.0
ive been trying to get this problem for the last like hour
 2 years ago

2bornot2bBest ResponseYou've already chosen the best response.1
Let \(y=mx+c\) be the tangent to the ellipse and let the point \((m,0) \) also lie on it
 2 years ago

satellite73Best ResponseYou've already chosen the best response.2
so we have to find the equation of the line through (m,0) that touches the ellipse right?
 2 years ago

satellite73Best ResponseYou've already chosen the best response.2
\[y=cxm\]since we cannot use m for the slope
 2 years ago

2bornot2bBest ResponseYou've already chosen the best response.1
Right. we can't use m, since its already up there
 2 years ago

satellite73Best ResponseYou've already chosen the best response.2
and we have to make sure that \[c=\frac{x}{7y}\] as well so perhaps we will get two equations
 2 years ago

2bornot2bBest ResponseYou've already chosen the best response.1
OK, so lets assume that the line \(y=kx+c\) is a tangent to the given ellipse
 2 years ago

2bornot2bBest ResponseYou've already chosen the best response.1
So we know c must be equal to \(\pm \sqrt{a^2k^2+b^2}\)
 2 years ago

2bornot2bBest ResponseYou've already chosen the best response.1
Where a and b are from the ellipse
 2 years ago

satellite73Best ResponseYou've already chosen the best response.2
you are way ahead of me. i only know that the line must look like \[y=c(xm)\] where c is the slope. i also know that we must have \[c=\frac{x}{7y}\]
 2 years ago

2bornot2bBest ResponseYou've already chosen the best response.1
Yes, you are right. And I was thinking of considering that at the end. Because we have two conditions to consider. First the line is a tangent And the second is what you are doing, i.e. the point lies on that line
 2 years ago

2bornot2bBest ResponseYou've already chosen the best response.1
So what I am stating up there, is the condition for the line \(y=kx+c\) to be a tangent to that ellipse
 2 years ago

2bornot2bBest ResponseYou've already chosen the best response.1
And I am considering the general ellipse, \(\huge \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)
 2 years ago

satellite73Best ResponseYou've already chosen the best response.2
ok i have this, tell me what you think
 2 years ago

satellite73Best ResponseYou've already chosen the best response.2
we know the line has slope \[\frac{x}{7y}\] and it passes through (m,0) meaning it has the equation \[y=\frac{x}{7y}(xm)\] now we get \[7y^2=x(xm)\] \[7y^2=x^2+xm\] \[x^2+7y^2=xm\] and we also know that on the ellipse \[x^2+7y^2=8\] making \[xm = 8\]
 2 years ago

2bornot2bBest ResponseYou've already chosen the best response.1
So the if the line y=kx+c is a tangent to \(\huge \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) then by theorem we know \[c=\pm \sqrt{a^2k^2+b^2}\]
 2 years ago

2bornot2bBest ResponseYou've already chosen the best response.1
So the equation of the line is \[y=kx+\pm \sqrt{a^2k^2+b^2}\]
 2 years ago

2bornot2bBest ResponseYou've already chosen the best response.1
Also since the line passes through the point (m,0) \[0=km\pm \sqrt{a^2k^2+b^2}\]
 2 years ago

2bornot2bBest ResponseYou've already chosen the best response.1
The above equation will give you two values of k, by solving the quadratic. Solve them, and put them in the actual equation to the st line, and you will get the two tangents
 2 years ago
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