）Make a vertical line of asymptote through a focus F of hyperbola x ^ 2 / A ^ 2 - y ^ 2 / b ^ 2 = 1 (a > b > 0), if the vertical foot is just in of (o is the origin) What is the eccentricity of the hyperbola on the vertical bisector

）Make a vertical line of asymptote through a focus F of hyperbola x ^ 2 / A ^ 2 - y ^ 2 / b ^ 2 = 1 (a > b > 0), if the vertical foot is just in of (o is the origin) What is the eccentricity of the hyperbola on the vertical bisector

Hyperbola x ^ 2 / A ^ 2 - y ^ 2 / b ^ 2 = 1 (a > b > 0)
Then the asymptote equation is y = ± x * B / A
According to the condition: the perpendicular foot is just on the vertical bisector of of (o is the origin),
The abscissa of F is FX = C / 2, where C is the abscissa of the focus
The ordinate is FY = ± x * B / a = ± C / 2 * B / A
Of = √ (FX ^ 2 + FY ^ 2) = C / 2 √ (1 + B ^ 2 / A ^ 2) = C / 2 * e, where e is the eccentricity of hyperbola
That is, e = 2 * of / C
According to the similar properties of right triangles:
Fx:OF=OF:c
That is, of ^ 2 = FX * C = C ^ 2 / 2
OF/c=1/√2
Eccentricity of final hyperbola
e=2*OF/c=2*1/√2=√2
If the asymptote equation of hyperbola is y = plus or minus 2 / 3x and the hyperbola passes through (3,4), then the equation of hyperbola with the center at the origin and the focus on the coordinate axis is?
Let the equation of hyperbola be y ^ 2 / (2a) ^ 2 - x ^ 2 / (3a) ^ 2 = 1; substituting the coordinates (3,4) into the above equation, we get 16 / (2a) ^ 2 - 9 / (3a) ^ 2 = 1; → 16 / 2 ^ 2 - 9 / 3 ^ 2 = a ^ 2; → a ^ 2 = 3
Find f (x) = under the root sign (x ^ 2_ The minimum value of (x ^ 2-4x + 8) under 2x + 2) + radical
1 + radical 5
Y = x ^ 2-2x + 2 under root sign + x ^ 2-4x + 8 under root sign
=Root ((x-1) ^ 2 + 1) + root ((X-2) ^ 2 + 4)
Geometric meaning: Y represents the sum of the distance from point P (x, 0) on the X axis to point a (1, 1) and to point B (2, 2).
Now is the minimum value of the sum of the two distances required!!!
P is on the X axis, not on ab
Let B be a point B 'symmetric about X axis, then Pb = PB'
PA + Pb = PA +... Expansion
Y = x ^ 2-2x + 2 under root sign + x ^ 2-4x + 8 under root sign
=Root ((x-1) ^ 2 + 1) + root ((X-2) ^ 2 + 4)
Geometric meaning: Y represents the sum of the distance from point P (x, 0) on the X axis to point a (1, 1) and to point B (2, 2).
Now is the minimum value of the sum of the two distances required!!!
P is on the X axis, not on ab
Let B be a point B 'symmetric about X axis, then Pb = PB'
PA+PB=PA+PB'.
It can be seen that when three points are collinear, the minimum distance is ab '
Ab '= root 10. The minimum value of the function y = x ^ 2-2x + 2 + x ^ 2-4x + 8 under the root sign is the root sign 10. Put it away
We do it geometrically
F (x) = √ (x ^ 2-2x + 2) + √ (x ^ 2-4x + 8)) = √ ((x-1) ^ 2 + 1) + √ ((X-2) ^ 2 + 4) is equivalent to finding the minimum distance from X (x, 0) to a (1,1) and B (2,2).
In the plane rectangular coordinate system, find the symmetric point a '(1, - 1) of A. if there is symmetry, we can know that the distance of | a'B | is the required distance.
The answer is the root 10
I just want to say that the second floor and the third floor are right
Inequality 2x ^ 2-11x + 12
(2x-3)(x-4)
The set form is (3 / 2,4)
The normal form is 3 / 2
Solving inequality: 2x ^ 2-11x + 12 > 0?
2X^2-11X+12>0
（2x-3）（x-4）>0
Therefore, 2x > 3 and x > 4, that is, x > 4
Or 2x
Given the function f (x) = x & # 178; - 2aX + 5, if the equation f (x) = 1 has a real root, find the value range of A
Given the function f (x) = x & # 178; - 2aX + 5, if the equation f (x) = 1 has a real root, the value range of a is obtained. Let a > 1, and the domain and range of F (x) are [1, a], the value of real number a is obtained
F (x) = 1 has real roots
X & # 178; - 2aX + 4 = 0 has real roots,
Thus, the discriminant 4A & # 178; - 16 ≥ 0
The solution is a ≥ 2 or a ≤ - 2
Solve the equation 100-3.6x = 64-2.4x 78 + 3.2X = 56 + 0.8x in ten minutes
100-3.6x=64-2.4x
100-64=-2.4x+3.6x
36=1.2x
x=30
78+3.2x=56+0.8x
78-56=0.8x-3.2x
22=-2.4x
x=-55/6
Given that the square root of a positive number is a + 3 and 2a-15, how much is the square root of this number
Give me the process! OK？
We know that the square root of a positive number is a + 3 and 2a-15,
Then a + 3 + 2a-15 = 0
A = 4
So, a + 3 = 7, 2a-15 = - 7
The square roots of this number are 7 and - 7
Because the square root of a positive number has two opposite numbers,
So a + 3 = 15-2a
The solution is a = 4
So the square root of this number is 4 + 3 = 7
Because the square roots of a positive number are opposite to each other
So a + 3 + 2a-15 = 0
A=4
4 + 3 = 7, so the square roots are 7 and - 7
It is known that the quadratic equation kx2-4kx + K-5 = 0 with one variable has two equal real roots. Find the value of K and the real roots of the equation
∵ the univariate quadratic equation kx2-4kx + K-5 = 0 has two equal real roots, ∵ K ≠ 0 and △ = 16k2-4k (K-5) = 0, ∵ 4K (3K + 5) = 0, the solution is k = - 53, ∵ the univariate quadratic equation about X is - 53x2 + 203x-203 = 0, ∵ x2-4x + 4 = 0, namely (X-2) 2 = 0, ∵ X-2 = 0, ∵ X1 = x2 = 2
How to solve the equation when 14 (X-5) = 56?
solution
x-5=56÷14
x-5=4
X=9