How many proofs for conics do we need to know? (1 Viewer)

[shady145]

hey just wondering how many we are expected to know.
and can some1 plz do this 1 for me. ty

the tangent at p, any point on a hyperbola, meets the directrix at T. show that PT subtends a right angle at the focus.

 

[Trebla]

All of them.

However, they wouldn't be likely to pop up in an exam...

 

[shady145]

All of them.

However, they wouldn't be likely to pop up in an exam...

YAY but i will do them just incase.

 

[Trebla]

the tangent at p, any point on a hyperbola, meets the directrix at T. show that PT subtends a right angle at the focus.

For the positive case P(x₁,y₁) for x > 0 and S(ae,0):
Equation of tangent at P (derive this)
xx₁ / a² - yy₁ / b² = 1
When x = a/e at the directrix
x₁ / ea - yy₁ / b² = 1
y = b²(x₁ - ea) / eay₁
T(a/e, b²(x₁ - ea) / eay₁)
m_PS = y₁ / (x₁ - ae)
m_ST = {b²(x₁ - ea) / eay₁} / (a/e - ae)
= b²(x₁ - ea) / (a²y₁ - a²e²y₁)
= a²(e² - 1)(x₁ - ae) / a²y₁(1 - e²)
= - (e² - 1)(x₁ - ae) / y₁(e² - 1)
= - (x₁ - ae) / y₁
m_PS x m_ST = [y₁ / (x₁ - ae)]. [- (x₁ - ae) / y₁] = -1
.: PT subtends a right angle at the focus

If P(x₁,y₁) for x < 0 and S(- ae,0) instead, then by symmetry the result should also hold for this case.

 

[shady145]

thank you,
i can see where i went wrong,
in the last part mutiplying the two gradients

 

[cutemouse]

All of them.

However, they wouldn't be likely to pop up in an exam...

Which proofs would they be, and how many are there?

 

[Trebla]

Which proofs would they be, and how many are there?

Refer to the syllabus...