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## Precalculus (2018 edition)

### Course: Precalculus (2018 edition)>Unit 2

Lesson 12: Challenging conic section problems (IIT JEE)

# Common tangent of circle & hyperbola (2 of 5)

2010 IIT JEE Paper 1 Problem 45 Circle Hyperbola Common Tangent Part 2. Created by Sal Khan.

## Want to join the conversation?

• I'm watching this video in the Algebra: Conic sections topic. What does IIT JEE mean?
• It is the toughest exam during under graduate level in the world. (Approved)
(1 vote)
• How does the IIT JEE compare to other Standardized tests in America, like the SAT or ACT?
• IIT JEE is a test that requires a lot of intelligence, practice, effort and time-management. SAT is just for kids. SAT requires taking excellent English classes in school, and practicing time-management. The SAT and IIT JEE are not exams that can even remotely be compared.

I took the IIT JEE in 2011 and got an All India Rank of 4249 (out of 5,00,000 candidates).

I also took the SAT with a week's preparation (only took practice tests at home a week before the exam) and scored 2100/2400. I haven't taken the SAT Subject Tests but seeing the sample questions, I feel even the Subject Tests are not anywhere close to the IIT JEE.

But I won't agree with Rishi (above); IIT JEE is NOT of the same standard as the IPhO or IChO. The Olympiads are tougher.
• what does IIT JEE mean?
• At , why isn't it 8(sqrt(m² + 64)) instead of 8(sqrt(m² + 1)) ? It looks like it should go from sqrt(8²m^2 + 4•16) to sqrt(m² + 64) • sqrt(8²) to 8(sqrt(m² + 64)). Thanks!
(1 vote)
• You need to factor 64 out of both terms.
sqrt(8²m^2 + 4•16) = sqrt(64m^2 + 64) = sqrt(64(m^2+1)) = sqrt(64)sqrt(m^2+1). Remember that square roots separate across multiplication, not addition.
• While Sal is doing the quadratic equation he assumes that b=8m, not 8mb, but in the quadratic from which he is deriving the quadratic equation the b term is 8mb. Why the omission of the b term?

Also, shouldn't the c term be -16, why does Sal write a positive 16?
• Sal Probably shouldn't have defined the y-intercept as b as the b in the quadratic equation is only the coefficient of b, and that is 8m. C is -16, but it got multiplied by -4, so it becomes positive.
• First of all, thank you for this video.

Big picture: I am confused about using math "tricks". In particular the "trick" Mr. Khan uses around of substituting an entire general equation into a specific equation.

Early in the video (around ) Sal, put a generic equation of a standard line (y-mx+b) into a SPECIFIC equation for a circle. To me this seems like math wizardry and I am VERY curious about receiving an answer on this. It appears to me you could throw any number of equations and play with them algebraically, how do you know which equation to use? and where to put it?

When Mr. Khan does the tricky "add 1 to both sides" (and 1 can be anything divided by itself).. I can follow that very well. But this is blowing my mind.

When can you use this "put a standard equation into a specific equation" trick?

As a side note, Mr. Khan also did some "put a standard equation into a specific equation" trick/wizardy in the previous video ("Common tangent of a circle & hyperbola (1 of 5)). I ignored it because he said something like "if you're taking this specific test, just remember this rule for time purposes". So I ignored it.

But this is the 2nd time he has used it. And I really don't understand when, why, and how to use this trick/wizardy/tool.

Summary: When, why and how can you substitute "standard" (y=mx+b) equations in specific equations for an object (circle, hyperbola, etc)?
• the answer to when can we substitute standard equation of a line in specific
equations:
when you will have to find an equation of either the tangent to the given object or the intersection points when suppose a line having general equation y=mx+c either intersects it or is a tangent to it
why and how:
it's just like solving simple quadratic equations !! whenever you are supposed to find intersection points then you will always have to solve ''simultaneously'' the given equations as sal did in the video he just simultaneously solved by substituting the equation of line i.e with degree 1 into another equation of degree 2
so the point is that in general to find the equation of tangent that is line (y=mx+c) or intersection points you have to use the standard equation and conics equation and simultaneously solve them
hope you get the point
Thank You.. !
• I understand why we set b^2-4ac to equal zero. What I'm wondering is what happens when we do that? Don't we mess up the algebra when something that might had not equaled zero now equals zero?
(1 vote)
• The whole equation becomes -b/2a, but that is allowed in algebra, so we're not messing anything up.