Conics (1 Viewer)

[Currybear]

I havs a math question;

The line y = mx+b is a tangent to the elipse at X^2/9+Y^2/4=1

Show that b^2=9M^2+4

I know you can do this using the discriminate and a shit load of algebra, but is there a way to do it via equation the coefficients of a general tangent of an elipse to y=mx+b?

 

[braintic]

I havs a math question;

The line y = mx+b is a tangent to the elipse at X^2/9+Y^2/4=1

Show that b^2=9M^2+4

I know you can do this using the discriminate and a shit load of algebra, but is there a way to do it via equation the coefficients of a general tangent of an elipse to y=mx+b?

I don't understand. By 'equation' do you mean 'equating'? You are not allowed to quote the equation of the tangent to an ellipse in an exam. If you derive the equation of the tangent first, then yes.

 

[Currybear]

*equating

our teacher said we were allowed to memorise equations, and my friends tutor who is a hsc marker

 

[asianese]

It's not that hard..lol The algebra isn't that difficult.
And as braintic said you can't quote it.

 

[Currybear]

but is it possible?

 

[Makematics]

but in our school exam w had to quote several equations, despite the fact that our teacher said they would always either give it to you or ask it to be proven. i was so pissed cos i hadnt memorised any via direct study. luckily i remembered them though -.-

 

[HeroicPandas]

I don't understand. By 'equation' do you mean 'equating'? You are not allowed to quote the equation of the tangent to an ellipse in an exam. If you derive the equation of the tangent first, then yes.

what??? i never knew this....

if u can quote it, here is my method:

I havs a math question;

The line y = mx+b is a tangent to the elipse at X^2/9+Y^2/4=1

Show that b^2=9M^2+4

I know you can do this using the discriminate and a shit load of algebra, but is there a way to do it via equation the coefficients of a general tangent of an elipse to y=mx+b?

Yes u can. It's called the method of Congruency (well this is what i call it)

Take out the general tangent formula of an ellipse:


Re-arrange the equation of the tangent y = mx +b so it looks like the general tangent of an ellipse:



Equate coefficients of x and y for equations (1) and (2)



(x1,y1) lies on ellipse,



Sub x_1 and y_1 into this and u get ur desired condition

NOTE: i've added extra steps in some parts of the algebra. It'll be beneficial to do most of the algebra in ur head