Conics question (1 Viewer)

[sasquatch]

A question im stuck on:

THe first part is: Show y = mx + k is a tangent to the hyperbola (x²/a²) - (y²/b²) = 1, then m²a² - b² = k².

I can do that one but the second part follows on from this,

Hence find the equations of the tangents from the point (1,3) to the hyperbola (x²/4) - (y²/15) = 1 and the coordinates of their points of contact.


If it says "hence" does that DEFINATELY mean to follow on from what i have proven. As i would be able to solve the second question by using the chord of contact formula, instead of what i just proved now...

 

[Trev]

Subbing (1,3) into the equation y=mx+k you get the equation 3-m=k.
With the hyperbola given, a²=4 and b²=15 which you can sub into m²a²-b²=k² to get 4m²-15=k². Using simultaneous equations:
4m²-15=9-6m+m²
(m-2)(m+4)=0
When m=2, k=1; when m=-4, k=7. So equations of the tangents are y=2x+1 and y=-4x+7.

 

[sasquatch]

woah..haha i didnt even think of subbing in the value of the point (1,3) into the equation y = mx + k :| i dunno whats wrong with me today... thanks!

 

[shimmerz_777]

we tried that exact same question in class today... filled the whole whiteboard with pointless facts before realising how easy it was.

 

[sasquatch]

man i hate it when that happens!! makes me feel so stupid..