Hyperbola question (1 Viewer)

[_ShiFTy_]

P(x1, y1) is a point on the hyperbola x^2/25 - y^2/16 = 1
Prove tangent..blah blah blah

a) tangent cuts at G...find G
done G(25/x1 , 0)

b) prove that SP/S'P = SG/S'G
Is there another way of doing this without using the distance formula..cos it takes 10 years

 

[SeDaTeD]

Prove that S'M', GM and SP are concurrent, where M and M' are the points on the directrices with the same ordinate as P. Then use geometric arguements.

Edit: Noticed an error in the above. Mark a point S'' |GS| units away from G, on the opposite side of S, and on the x-axis. Now it's show the lines S'M', G'M and GP are concurrent. I tried with the above and it didn't work, because I forgot that P divided M'M externally whereas G divided SS' internally. The new point S'' now makes G divide S'S'' externally in the ratio as P divides MM'. It is that one you should use.

 

[haboozin]

you better remember what sedated said carefully, it will save you lots. Its very easy and it could gain u easy marks (while others dont even attempt because they are doing the distance formula)

 

[littleboy]

can you gimme an example plz??

 

[Dimsimmer]

lol shifty, i think that other thread that i started probably gave you the idea to start this thread. Anyways, i reckon that its good to find quicker ways to do conics questions.

 

[_ShiFTy_]

I dont quite understand Sedated's method, can someone draw up a diagram please?

 

[SeDaTeD]

Here are some diagrams: (note I have edited what I posted above)

 

[SeDaTeD]

Actually, I think the quickest method would just be:
SP/S'P = ePM/ePM' = PM/PM', which is easy to find since P, M and M' all lie on the same horizontal line. Then show that it's equal to SG/S'G, which is also easy to find since they're both on the x-axis.