Y axis intercepts of ellipse tangents

[Appleton]

Homework Statement

It is given that the line y= mx + c is a tangent to the ellipse

<br /> \frac{x^2}{a^2} + \frac{y^2}{b^2}=1 if a^2m^2=c^2-b^2

Show that if the line y=mx+c passes through the point (5/4, 5) and is tangent to the ellipse 8x^2+3y^2=35, then c = 35/3 or 35/9

Homework Equations

The Attempt at a Solution

The tangents should be in the form
y=\pm \frac{c^2-b^2}{a^2}x+c
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

<br /> y^2-2cy+c^2=\frac{c^2-b^2}{a^2}x^2\\<br /> c=\frac{(x(\pm \sqrt{a^2y^2-a^2b^2+b^2x^2}))+ay^2}{a^2-x^2}<br />

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?

 

[mfb]

How do you multiply tangent equations together and what do you expect as result?

You know a and b, so you get an equation involving m and c and known constants only.
The point (5/4, 5) gives another equation with m and c only. This one is linear, so I would use this and plug it into the other one. Ideally, you get the right two solutions for c.

7.719285513 and 3.827106245 are not 35/3 and 35/9.

 

[Appleton]

How do you multiply tangent equations together and what do you expect as result?

You know a and b, so you get an equation involving m and c and known constants only.
The point (5/4, 5) gives another equation with m and c only. This one is linear, so I would use this and plug it into the other one. Ideally, you get the right two solutions for c.

7.719285513 and 3.827106245 are not 35/3 and 35/9.

Hi mfb, thanks for your reply. Do you mean that I should plug m=4-\frac{4}{5}c into \frac{35}{3}m^2=c^2-\frac{35}{8}?

When I do this I get values for c identical to those I had previously (although perhaps a bit more efficiently this time):

\frac{35}{3}(4-\frac{4}{5}c)^2=c^2-\frac{35}{8}\\<br /> (c-\frac{560}{97})^2=\frac{313600}{9409}-\frac{68775}{2328}<br />
c=7.719285508 or 3.827106245
This suggests to me that multiplying the tangent equations was legitimate if a little unnecessary. The aim was to eliminate the square root, which your method bypasses quite successfully.

My main question now is should I bin my book. Are the values for c stated in the question correct?

Sorry if I've misinterpreted your guidance, I'm not ruling out this possibility.

 

[mfb]

I think you swapped the values for a and b.

 

[Appleton]

Isn't the convention to assign a to the value of the semi major axis? In this case 35/3 > 35/8 so shouldn't a=35/3 and b=35/8?

To check I'm not mistaken I repeated the procedure above, swapping a and b, and ended up with the square root of a negative number.

 

[mfb]

The order of a and b does not matter, but it has to be consistent within the problem. The problem divides x^2 by a^2, so you cannot call that b^2.

WolframAlpha finds nice solutions - exactly those given.

 

[Appleton]

Looks like the negative square root i produced above was an error. Having gone through it again I am able to produce the values for c in the question. Thank you for identifying my error in mixing up a and b.

 

[TheMathNoob]

Homework Statement

It is given that the line y= mx + c is a tangent to the ellipse

<br /> \frac{x^2}{a^2} + \frac{y^2}{b^2}=1 if a^2m^2=c^2-b^2

Show that if the line y=mx+c passes through the point (5/4, 5) and is tangent to the ellipse 8x^2+3y^2=35, then c = 35/3 or 35/9

Homework Equations

The Attempt at a Solution

The tangents should be in the form
y=\pm \frac{c^2-b^2}{a^2}x+c
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

<br /> y^2-2cy+c^2=\frac{c^2-b^2}{a^2}x^2\\<br /> c=\frac{(x(\pm \sqrt{a^2y^2-a^2b^2+b^2x^2}))+ay^2}{a^2-x^2}<br />

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?

I am just curious. For which class is this?, Real analysis?

 

[Appleton]

I am just curious. For which class is this?, Real analysis?

The chapter of the book is called coordinate geometry. I'm not sure if there is a more appropriate title for this branch of maths.