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It is known that the asymptote equation of hyperbola C is y = ± 3x, and the distance from the right focus f (C, 0) to the asymptote is 3. (1) find the equation of hyperbola C; (2) make a straight line with slope k through F, l intersects the hyperbola at two points a and B, and the central perpendicular of line AB intersects the X axis at D
(1) Let the hyperbolic equation be 3x2-y2 = λ (λ > 0) (2 points) from the question, we know that C = 2, ∧ λ 3 + λ = 4, ∧ λ = 3 The hyperbolic equation is: x2 − Y23 = 1 (5 points) (2) let the equation of the line l be y = K (X-2) and substitute it with x2 − Y23 = 1 to get (3-k2) x2 + 4k2x-4k2-3 = 0 (6 points) let a (x1, Y1), B (X2, Y2), the midpoint P (x0, Y0) of AB, then x0 = - 2k23 − K2, substituting l to get: Y0 = - 6k3 − K2 (7 points) | ab | = 1 + K2 | x1 − x2 | = ＝6(k2+1)|3−k2|… The vertical bisector equation of AB is y = - 1K (x + 2k23 − K2) − 6k3 − K2 Let y = 0 get XD = - 8k23 − K2 (10 points) | FD | = | - 8k23 − K2 − 2 | = | - 6 (1 + K2) 3 − K2 | = 6 (1 + K2) | - 3 − K2 | (11 points) | ab | FD | = 1 is the fixed value (12 points)
Find the standard equation of hyperbola which is asymptote with hyperbola x ^ 2-y ^ 2 / 2 = 1 and focus with ellipse x ^ 2 / 2 + y ^ 2 = 1
Because the hyperbola and x ^ 2-y ^ 2 / 2 = 1 share the same asymptote, let x ^ 2-y ^ 2 / 2 = K be changed to x ^ 2 / K-Y ^ 2 / (2k) = 1. In the ellipse x ^ 2 / 2 + y ^ 2 = 1, a ^ 2 = 2, B ^ 2 = 1, so C ^ 2 = a ^ 2-B ^ 2 = 1, then the focus is (- 1,0), (1,0). Therefore, from the known K + 2K
It is known that the focus of hyperbola coincides with the focus of ellipse x ^ 2 / 16 + y ^ 2 / 12 = 1, and the asymptote equation is y = ± X
ellipse
c'²=a'²-b'²=4
c'=2
Then hyperbola C = 2
Focus (± 2,0)
Asymptote slope ± B / a = ± 1
So B = a
a²+b²=c²=4
So a & sup2; = B & sup2; = 2
x²/2-y²/2=1
A1 / 2 = B4 / 5
So a: B = 4 / 5:1 / 2 = 8:5
That is, a is 8 / 5 of B
So 1 / 3 of a is 8 / 5 × 1 / 3 of B = 8 / 15
That is, 8 / 15 of B is 2 times smaller than B
So B is 2 (1-8 / 15) = 30 / 7
A is 30 / 7 × 8 / 5 = 48 / 7
30/7+48/7=78/7
So a and B are 78 / 7
Let the two focal points of hyperbola Y2 / a2-x2 / 3 = 1 be F1F2 and its eccentricity be 2. Find two quasilinear equations
E = C '/ a' = 2
c'=2a'
c'²=4a'²
Here a '& sup2; = 3
So c '& sup2; = 12
c'=2√3
Therefore, the guide line x = ± a '& sup2 / C' = ± √ 3 / 2
It can be seen from the meaning of the title:
bˇ2=3
e=c/a=2
c=2a cˇ2=4aˇ2
cˇ2=aˇ2+bˇ2
So 4A ˇ 2 = a ˇ 2 + 3
A=1
C=2
The guide line is y = ± √ 3 / 3 X
Verification: formula of summation of equal ratio sequence
How to prove the sum formula of equal ratio sequence!
1. Define A2 = A1 * q, A3 = A2 * q... a (n-1) = a (n-2) * q, an = a (n-1) * q by the equal ratio sequence, and add n-1 equality sides to get A2 + a3 +... + an = [A1 + A2 +... + A (n-1)] * q, that is, sn-a1 = (SN an) * q, that is, (1-Q) Sn = A1 an * q, when Q ≠ 1, Sn = (A1 an * q) / (1-Q) (n ≥ 2)
Let the equal ratio sequence an and the ratio be I
Then A2 = A1 * I,
a3=a2*i=a1*i^2
......
an=a1*i^(n-1)
Sequence and Sn = a1 + A2 + a3 +... + an
Sn*i=a2+a3+....+an+an*i
Sn*i-Sn=an*i-a1=a1*[i^n-1]
Sn=a1*[i^n-1]/(i-1)
q=1:Sn=n*a1
Q is not equal to 1: subtracting by dislocation: SN = a1 + A2 + a3 +... + an --- 1
qSn= a1q+a2q+... + an-1q+anq ----2
(1-Q) Sn = A1 anq = A1 (1-Q ^ n)
Sn=....
The fourth grade of primary school mathematics exercise book the problem of equation class
What question? Can you send it?
In a special sense, the theory of length, the theory of relativity Is the formula for an object moving at high speed or for an observer in a static reference frame?
It is the change of the reference frame caused by the moving object
In the triangle ABC, sinbsinc = the square of Cos2 / A, find the shape of ABC
The summation formula of arithmetic sequence is (first term + last term) / 2 * number of terms. How to find the number of terms?
For example, 2 58 11 14 ·······································································································································
(last first) / tolerance + 1
What are the four numbers represented by "cultural primary school"?
* 4
—————
Learning small culture
* 4
————
Learn xiaohuawen
2178×4＝8712
RELATED INFORMATIONS
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