conics : mx + c, for touching (1 Viewer)

[underthesun]

this is also for proving tangents, but

do you think in exam situations, they'll ever ask these kinds of questions?

"Prove that the condition for the line y = mx + c to be a tangent to the ellipse (equation goes here) is c^2 = a^2 m^2 + b^2"

terry lee book does it the long algebra line way, excel book does a relatively short "comparing co-efficients" way, but it requires that you have the tangent..

somebody enlighten me?

 

[turtle_2468]

I haven't done conics for a while, but this is an ok way of doing it..
another way could be implicit differentiation for the ellipse. So you get x^2/a^2+y^2/b^2=c, or something like that. Shift y term to the other side and you see that you can implicitly differentiate to get dy/dx = some function of x and y. Then if you know the gradient of the curve, sub in here to get a relation of x and y, then sub that back into the ellipse for possible x and y values (note that there are 2 solns). That could be a shorter way of doing it... try it, it only takes a few minutes!

 

[OLDMAN]

I do not have the Excel book, but could imagine the approach. It is a good short approach, if all that is asked is whether a line touches a curve, and you know the form of the tangent equation eg.
xX_1/a^2+-yY_1/b^2=1. However, if the question asks whether the line misses or cuts the curve, Terry Lee's approach is more appropriate. It looks long and messy, but isn't really, try it a few times and you'll find things fall into place.
I can't see any reason why this type of question couldn't be asked in an exam, after all you would have been introduced to discriminants even in 2 Unit.

 

[underthesun]

thanks

I'm off practicing the T.Lee methods now, three times until I can do it perfectly..

 

[gronkboyslim]

use the parametric way

 

[underthesun]

Originally posted by gronkboyslim
use the parametric way

do you mean comparing co-efficients?