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Course: Algebra (all content)>Unit 17

Lesson 12: Challenging conic section problems (IIT JEE)

Representing a line tangent to a hyperbola

How a tangent line relates to a hyperbola. Might be useful for some competitive exams where there isn't time to derive (like we are doing in this video). Created by Sal Khan.

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• I'm currently in Calc 2 and we worked on conic sections.....so how is conic sections applicable to Calc 2?
• Conic sections provide a rich zoo of possible curves and scenarios for use in many methods learned in calculus. I don't know which Calculus 2 curriculum you are in, so I cannot give an exact mapping. Some schools offer Calculus over 3 Semesters (or more), while others offer it over 4 Quarters, or a more intensive series in 2 Quarters. Calculus for non Math/Physics Majors is different from calculus for Math and Physics majors. But anyway, across all the topics of Calculus, conics keep popping up: just think of solids of revolution...surface areas and volumes...parametric representation of curves...finding slopes of tangents to whatever conic...rotating axes of curves for conics...parametrics...particle motion...and so on.
Yum. Knowing the various relationships of conic sections gives you avenues into setting up otherwise impossible problems for successful solution.
• In the video Sal says he did conic sections IIT JEE problems in previous videos, but this video is the first in the series. Where can I find the previous videos?
• There are many IIT videos here: https://www.khanacademy.org/test-prep/iit-jee-subject/iit-jee - several of them involve conic sections, so he's presumably referring to those.

(It's weird, those are off in their own little "Test prep" category, instead of being in the "Math" category. Not sure what the purpose is of hiding them over there. Seems like they should be moved here.)
• How many questions are in the exam? And subsequently how much time?
• It says that the video is, "very optional" so will this be on a precalculus final or not?
• This might sound like a basic question, but what does IIT JEE mean? What does it stand for?
• Indian Institutes of Technology- Joint Entrance Exam
• is this relationship that Sal found in the end "" is true also for a hyperbola opens up and down ??
• At , what if the coefficient of the x^2 term equals zero? Then, m^2-b^2/a^2 = 0 , and it will become a liner equation which it still has only one solution.
I hope my question is clear enough :-)
• Well, then line has an equation y = b/a*x + c or
y = -b/a*x + c. This lines are asymptotes of hyperbola shifted up(down) by c units. They intersect hyperbola in only one point, but they are not tangents. I wonder myself, why this case was missed in the video?
• What would be the tangent line relation for the hyperbola (y^2/a^2) - (x^2/b^2) = 1 ?
Would it be x = my + c where c^2 + b^2 = (a*m)^2 ??
What would be the relation for the hyperbola xy = a constant ?
• Why did sal stop when he had the hyperbola equation equal to y^2? Why didn't he just take the square root of both sides to get y=b/a*x-b?
• Maybe because the square root of [b^2x^2/a^2 - b^2] does not create b/a*x-b
You can not split the square root across terms. It just won't work.
For example: sqrt(16+9) = sqrt(25) = 5
If you use your methoed: sqrt(16+9) = sqrt(16) + sqrt(9) = 4 + 3 = 7. This is an incorrect value.
Hope this helps.
(1 vote)
• At , how does Sal arrive at (m^2-b^2/a^2) x^2? What happened to the x^2 term that was part of b^2/a^2?
(1 vote)
• It was not lost. Sal didn't write out all the steps. Here's basically what happened...
Sal started with: m^2x^2 - (b^2x^2)/a^2
He has 2 terms that contain a common factor of x^2. He used the distributive property to factor out the x^2. This creates: m^2-b^2/a^2) x^2

Hope this helps.