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Let △ ABC be an isosceles triangle, ∠ ABC = 120 °, then the eccentricity of the hyperbola with a, B as the focus and passing through point C is () A. 1+22B. 1+32C. 1+2D. 1+3
From the meaning 2C = | ab |, so | AC | = 2 × 2C × sin600 = 23C, from the definition of hyperbola, there are 2A = | AC | - | BC | = 23c-2c {a = (3-1) C, | e = CA = 13-1 = 1 + 32, so choose B
A hyperbolic high school problem
7. Let F1, F2, and be the left and right focal points of the hyperbola x2 / a2-y2 / B2 = 1 respectively. If there is a point P on the right branch of the hyperbola, satisfying | Pf1 | = | F1F2 |, and the distance from F2 to the straight line Pf1 is equal to the real axis length of the hyperbola, then the asymptote equation of the hyperbola is
|Pf1 | = | F1F2 | = 2C | Pf1 | - | PF2 | = 2A | PF2 | = 2c-2a triangle pf1f2 is an isosceles triangle. The distance between PF2 and straight line Pf1 is equal to the real axis length 2a of hyperbola
If there is no common point between hyperbola y ^ 2 / A ^ 2-x ^ 2 / b ^ 2 = 1 and circle x ^ 2 + y ^ 2-10x + 20 = 0, then the value range of hyperbolic eccentricity is?
a>0,b>0
The circular equation is: x ^ 2 + y ^ 2-10x + 20 = 0, which becomes (X-5) ^ 2 + y ^ 2 = (√ 5) ^ 2,
Then the equation is a circle with (5,0) as the center and √ 5 as the radius,
If there is no common point between the hyperbola and the circle, then the two asymptotes of the hyperbola are tangent lines of the circle at the limit position,
Let two tangent equations passing through the center of a circle be y = ± KX,
Let's look at one, y = KX,
According to the formula of point line distance, d = | 5 * k-0 * 1 | / √ (1 + K ^ 2) = √ 5,
|k|=1/2,
|The minimum K | is 1 / 2,
The tangent equation is an asymptote equation of hyperbola, y = ± (A / b) X,
Then a / b = | K |,
b=a/|k|,
c=√（a^2+a^2/k^2)=a√(1+k^2)/|k|,
Eccentricity e = C / a = √ (1 + K ^ 2) / |k | = √ (1 + 1 / K ^ 2)
|The larger K | is, the smaller e is. K →∞, e → 1, (close to Y-axis),
On the contrary, the smaller the | K | is, the larger the fraction e is. When it reaches the limit position and reaches the tangent line of the circle, that is, when | K | = 1 / 2, the tangent line of the circle is the asymptote of the hyperbola, e = √ 5. Although it is tangent to the circle, it does not intersect the hyperbola,
∴1
If the distance from point P to point (5,0) on hyperbola x216-y29 = 1 is 15, then the distance from point P to point (- 5,0) is ()
A. 7b. 23C. 5 or 25d. 7 or 23
∵ hyperbola x216-y29 = 1, ∵ 2A = 8, (5,0) (- 5,0) are two focal points, ∵ point P is on hyperbola, ∵ Pf1 | - | PF2 | = 8, ∵ the distance from point P to point (5,0) is 15, then point P to point (- 5,0) is 15 + 8 = 23 or 15-8 = 7, so D
Hyperbola in Senior High School
Given that the center of the hyperbola is at the origin, cross the right focus f (2,0) to make a straight line under the root of the slope (3 / 5), intersect the curve with two points m and N, and | Mn | = 4, solve the hyperbolic equation
I just can't figure out what a ^ 2 B ^ 2 is,
I hope someone can calculate it for me, and the calculation process won't be necessary,
y=√(3/5)*(x-2)
Let the hyperbolic equation be x ^ 2 / A ^ 2-y ^ 2 / b ^ 2 = 1 m (x1, Y1) n (X2, Y2)
The simultaneous equation is (5b ^ 2-3a ^ 2) x ^ 2 + 12a ^ 2x-12a ^ 2-5A ^ 2B ^ 2 = 0
x1+x2=-12a^2/(5b^2-3a^2),x1*x2=(-12a^2-5a^2b^2)/(5b^2-3a^2)
A ^ 2 + B ^ 2 = 4, B ^ 2 = 4-A ^ 2 is substituted into X1 + X2, X1 * X2, so there is only formula about a ^ 2 in the two formulas
The chord length formula √ (1 + K ^ 2) [(x1 + x2) ^ 2 - 4x1x2] = 4 is used to solve a ^ 2,
So we can get B ^ 2, the hyperbolic equation is OK
Several high school mathematical problems about ellipse and hyperbola
Question 1: if f, F 1 (0,5) and F 2 (0, - 5) satisfy | Pf1 | - | PF2 | = 6, then the trajectory equation of point P is? Why is the answer to this question y ^ 2 / 9-x ^ 2 / 16 = 1 (Y ≤ - 3) I don't understand why y ≤ - 3 and how to take the value. Question 2: if x ^ 2 + KY ^ 2 = 2 represents an ellipse with focus on the Y axis, then the value range of the real number k is? A (0, + ∝), B (0,2), C (1,2), +How to find it? How to use it. Question 3: points F1 and F2 are the two focuses of the ellipse x ^ 2 / 9 + y ^ 2 / 7 = 1, a is the point on the ellipse, and ∠ af1f2 = 45 degrees, then the area of △ af1f2 is? A 7 b 7 / 4 C 7 / 2 D 7 √ 5 / 2
Because he wants to satisfy | Pf1 | - | PF2 | = 6, not - 6 pairs. The curve in the half is not good. As for why it is - 3, it's very simple. The highest point of the hyperbola in the lower half is - 3
I don't remember the second question. In a word, look at the size of 1 and 1 / K. you can read the book. The tree will tell you which fight and which axis the focus will fall on
In question 3, you can work out the equation of the straight line AF1, because it is 45 degrees. You don't need to consider too many cases. If you calculate the slope as 1, you can work out the height and area of the triangle by finding a suitable focus
Mathematical problems of ellipse and hyperbola
1： The standard equation for finding the ellipse with the length of long axis 12, eccentricity e = 1 / 3 and focus on Y axis
2： Find the standard hyperbolic equation suitable for the following conditions
1) The length of the imaginary axis is 2, passing through the point (√ 3,0), and the focus is on the X axis;
2) The focal length of the hyperbola is the length of the long axis of the ellipse X & # 178 / 25 + Y & # 178 / 9 = 1, and half of the real axis is 2 √ 3
The above / is equivalent to the fraction line, √ is the root sign
It's urgent!
1： The standard equation for finding ellipse with long axis 12, eccentricity e = 1 / 3, focus on Y axis 2A = 12, a = 6e = C / a = 1 / 3, C = 2B ^ 2 = a ^ 2-C ^ 2 = 36-4 = 32, focus on Y axis, then the equation is y ^ 2 / 36 + x ^ 2 / 32 = 1. 2: find the standard equation for hyperbola suitable for the following conditions
A mathematical problem (about ellipse and hyperbola)
If the distance between a point P (a, b) on the right branch of hyperbola X & # 178; - Y & # 178; = 1 and the straight line L: y = x is d = √ 2, then a + B =?
A * A-B * b = 1, | A-B | / √ 2 = √ 2, | A-B | = 2, a * A-B * b = 1 [a + b] [A-B] = 1, a + B = 1 / 2 or - 1 / 2
A mathematical problem about ellipse and hyperbola
The square of ellipse 34 / x + y of N and hyperbola n have the same focus. Then the value of real number n is? (please write the procedure)
The square of X / 16 / 16 of y of hyperbola n = 1
The focus is on the x-axis and C ^ 2 = n ^ 2 + 16
The square of ellipse 34 parts x + the square of ellipse n parts y = 1 C ^ 2 = 34-n ^ 2
Together,
c=5,N=3
If the ellipse equation is x ^ 2 / 4 + y ^ 2 = 1, passing through point B (- 1,0), can a straight line l be made, so that l intersects with ellipse C and points m and N, and a circle with diameter Mn passes through coordinate origin o? If it exists, the l equation can be obtained
The equation is l, and the slope is k, K, y = K (x + 1) and the equation is l, and the slope is k, y = K (x + 1), so (4K \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\35;178; (x1 + x2) + K & #17
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