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Appleton
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Homework Statement
It is given that the line y= mx + c is a tangent to the ellipse
[itex]
\frac{x^2}{a^2} + \frac{y^2}{b^2}=1[/itex] if [itex]a^2m^2=c^2-b^2[/itex]
Show that if the line y=mx+c passes through the point (5/4, 5) and is tangent to the ellipse [itex]8x^2+3y^2=35[/itex], then c = 35/3 or 35/9
Homework Equations
The Attempt at a Solution
The tangents should be in the form
[itex]y=\pm \frac{c^2-b^2}{a^2}x+c[/itex]
I tried substituting this value for y into the equation of the given ellipse but it became a bit messy so I deployed a different tact:
The intersection of the tangents is equal to (5/4, 5), so multiplying the tangent equations together should give the point of intersection enabling me to solve for c
[itex]
y^2-2cy+c^2=\frac{c^2-b^2}{a^2}x^2\\
c=\frac{(x(\pm \sqrt{a^2y^2-a^2b^2+b^2x^2}))+ay^2}{a^2-x^2}
[/itex]
Substituting a=35/3 and b=35/8 gives 7.719285513 and 3.827106245.
When I insert these values into a graph drawing app they seem approximately correct whereas one of the given values in the question, 35/3, does not as far I can tell.
Are the values for c given in the question correct. Are my values for c correct?