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If (A & # 178; - 3a-2) + a of y = (A-4) x is a quadratic function, find the value of (1) a and the relation of (2) function
(A & # 178; - 3a-2) + a of y = (A-4) x is a quadratic function, so (A & # 178; - 3a-2) = 2
a^2-3a-4=0 (a-4)(a+1)=0 a-4≠0
a=-1
The relation of function is as follows
y=-5x^2-1
The ellipse and hyperbola have the same focus F1 (- 4,0), F2 (4,0). Let ﹣ 1 and ﹣ 2 be the eccentricity of the ellipse and hyperbola respectively, and ﹣ 1 / ﹣ 2 = 1 / 4, the trajectory equation of the common point of the ellipse and hyperbola is obtained
C = 4E1 = 4 / A1, E2 = 4 / A2, from E1 / E2 = 1 / 4, A1 / A2 = 4, that is, A1 ^ 2 = 16a2 ^ 2B1 ^ 2 = A1 ^ 2-16 = 16 (A2 ^ 2-1) B2 ^ 2 = 16-a2 ^ 2 ellipse; X ^ 2 / 16a2 ^ 2 + y ^ 2 / 16 (A2 ^ 2-1) = 1 hyperbola; X ^ 2 / A2 ^ 2-y ^ 2 / (16-a2 ^ 2) = 1, even if two curves contain only one parameter A2, let the common point be (x, y), then only
It's so hard
Because the focus is the same, C1 and C2 in the elliptic and hyperbolic standard equations are the same. From the ratio of eccentricity, A2 / A1 = 1 / 4 can be obtained. In this way, we can write a group of G (x, y, A1) = 0, f (x, y, A2) = 0. A2 is represented by X or Y, and then we can get the trajectory equation
The teacher taught us something similar. I have it in my notes. Forget it, it's just
According to the corresponding values of the independent variable X of the quadratic function y = ax & # 178; + BX + C and the function y in the table below, we can see that the root of the quadratic equation AX & # 178; + BX + C = 0 with respect to X is
x …… -1 0 1 2 ……
y …… 0 -3 -4 -3……
When x = 0 and 2, the value of the function is the same, so the axis of symmetry is x = 1
When x = - 1, f (- 1) = 0, so one root is - 1, so the other root is 1 + [1 - (- 1)] = 3
So the root of the equation is - 1,3
-1 and 3. Because it's symmetric about x = 1.
When x = 0 and x = 2, y = - 3
It shows that x = 1 is the symmetry axis of quadratic function image
When y = 0, the value of X is the root of the equation
According to the table, x = - 1 is one of the roots
According to the axis of symmetry, the other root is x = 3
I hope my answer will help you. Take it_ ∩)O！
The relationship between the size and shape of hyperbolic eccentricity
B / a = root sign (e ^ 2-1), so the larger e is, the steeper the asymptote of hyperbola is, and the thinner the hyperbola is
It is known that the coordinates of the intersection point of the image of the first-order function y = KX + B and the inverse scale function y = K / X are (2,3), and the analytic expressions of the two functions are obtained
Let x = 2, y = 3 generations, y = K / X get: k = xy = 2 * 3 = 6
Let x = 2, y = 3, k = 6 generations, y = KX + B get: 3 = 2 * 6 + B, B = - 9
The analytic expressions of these two functions
y=6x-9
y=6/x
What is the formula of the length (focal radius) of the line segment connecting a point on a conic (including ellipse, hyperbola and parabola) with the corresponding focus in a conic
Let P (X., Y.) be a point on conic, then R (left) = a + ex.; R (right) = a-ex
The proof is as follows:
The focus is on the X axis
The Quasilinear equation L: x = A & # / C, or x = - A & # / C,
Focus f (C, 0), or F (- C, 0), P (X., Y.)
According to the second definition: the distance r to the right focus F of P, divided by the distance d from P to the right collimator L, is equal to the eccentricity e (E = C / a)
That is e = R / d
e=r/[(a²/c)-x.]
The reduction is: R (right) = a-ex
Similarly: R (left) = a + ex
The inverse proportion function y = - KX (k is not equal to 0) at x = 1, when the independent variable increases by 2, the value of function decreases by 2 / 3, then K
Substituting x = 1
y=-k
Substituting x = 3, y = - K
-k-2/3=3
And then do it yourself
-k*1=-k
-k*3=-k-2/3
And then it's solved
k=1/3
The inverse proportion function is in the form of y = K / x, not this one
What is eccentricity? Is it applied to ellipse or hyperbola?
E = C / a an ellipse is the same as a hyperbola
Given an inverse scale function y = k: X (k is not equal to zero), when the independent variable is - 3, the function value is 4, then what is the scale coefficient K of the inverse scale function
The first question of the first side of the second volume of the ninth grade of junior high school
xy=k.k=-12
Then k = 4 * 3 = - 12
4=k/(-3)
So k = - 4 / 3
What is the hyperbolic eccentricity formula
E = C / a = distance from point to focus / distance to guide line
RELATED INFORMATIONS
- 1. The value range of the independent variable in the function y = radical (x + 2) / X clearly indicates that the coefficient K of the inverse proportional function is not equal to 0. Why is the answer x > = - 2 and X ≠ 0 Value range of independent variable in function y = radical (x + 2) / X Why is the answer x > = - 2 and X ≠ 0 not x > - 2 and X ≠ 0
- 2. The image of a certain function passes through the point (- 1,2), and the value of function y decreases with the increase of independent variable x. please write a function relation that meets the above conditions___ .
- 3. If (X & # 178; - 2x + 1) of function f (x) = a is a decreasing function on (1,3), then the solution set of inequality log a (x + 1) > 0 about X is () A.{x|x>-1} B.{x|x>0} C.{x|x
- 4. F (x) is an odd function defined on R, when x > = 0 is f (x) = x ^ 2, if for any x belonging to t to t + 2, the inequality f (x + T) > = 2F (x) is constant, the value range of T is obtained
- 5. It is known that, as shown in the figure: in the plane rectangular coordinate system, O is the coordinate origin, the quadrilateral oabc is a rectangle, the coordinates of points a and C are a (10,0), C (0,4), point D is the midpoint of OA, and point P moves on the edge of BC. When △ ODP is an isosceles triangle with waist length of 5, the coordinate of point P is______ .
- 6. In the plane rectangular coordinate system, O is the coordinate origin, oabc is a rectangle, point a coordinates (10,0) point C coordinates (0,4) point D is the midpoint of OA, point P is on the BC side When the triangle ODP is an isosceles triangle with waist length 5, the coordinate of point P is?
- 7. As shown in the figure, in the right angle trapezoid oabc, the coordinates of OA ‖ CB, a and B are a (15, 0), B (10, 12) respectively. The moving points P and Q start from O and B respectively. Point P moves along OA to terminal a at a speed of 2 units per second, and point Q moves along BC to C at a speed of 1 unit per second. When point P stops moving, point Q stops moving at the same time. The line segments OB and PQ intersect at point D, and make de ‖ when passing point D Let the motion time of PQ be t (in seconds). (1) when t is the value, the quadrilateral pabq is isosceles trapezoid, please write out the reasoning process; (2) when t = 2 seconds, calculate the area of trapezoid ofBC; (3) when t is the value, △ PQF is isosceles triangle? Please write down the reasoning process
- 8. It is known that the eccentricity of ellipse C: x2 / A2 + Y2 / B2 = 1 is root sign 3 / 2, the straight line X-Y + 1 = 0 passes through the upper vertex of ellipse C, the straight line x = - 1 intersects the ellipse at two points a and B, P is different from the ellipse at any point a and B, the straight line AP BP intersects the straight line L: x = - 4 at two points Q, R, 1 respectively, to solve the equation of ellipse C, 2 to prove that the vector OQ × vector or is a fixed value
- 9. If the two focuses and the two vertices of the minor axis of the ellipse are the four vertices of a diamond with a 60 ° angle, then the eccentricity of the ellipse is 0 A. 1 / 2 b. √ 3 / 2 C. √ 3 / 3 d.1/2 or √ 3 / 2
- 10. In the plane rectangular coordinate system, the vertex o of the rectangular oacb is at the origin of the coordinate, the vertices a and B are respectively on the positive half axis of the x-axis, OA = 3, OB = 4, and D is the center of the edge ob In the plane rectangular coordinate system, the vertex o of the rectangular oacb is at the origin of the coordinate, the vertices a and B are respectively on the positive half axis of the X axis, OA = 3, OB = 4, and D is the midpoint of the edge ob. If e and F are two moving points on the edge OA and EF = 2, then the minimum perimeter of the quadrilateral cdef is the coordinates of points E and F
- 11. Since the independent variable x ≠ 0 and the function value y ≠ 0 in the inverse scale function y = K / X (K ≠ 0), its image is at the intersection of X axis and Y axis, that is, the two branches of hyperbola are infinitely close to 2, but never reach the coordinate axis. In addition, the two branches of hyperbola are symmetrical about coordinate 3, and about straight line y = x and straight line y = - x 4
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- 13. If the mean value + 1 of the absolute value power of K of polynomial 4xy - 1 / 5 (K-2) y is a cubic binomial, then K=
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- 15. The image of X-2 power + 1 (a > 0, a is not equal to 1) of function y = a must pass through the point?
- 16. If the x value of quadratic function is not equal to zero, how to draw (or mark) its image?
- 17. Drawing function image with absolute value （1）y=|x-5|+|x+3| (2) Y = 2x-3 x ∈ Z | x | contained in Z （3）y=x²-2|x|-1 （4）y=｛x²+2x（x≥0） -x²-2x（x＜0） The best way to draw these four functions is to teach me. Thank you
- 18. Given the function y = 3 / (4x + 8), then the value of the independent variable x is?
- 19. Given that the image of the function f (x) = ax & # 178; + BX + C passes through the origin, for any x ∈ R, f (x-1) = f (x + 1) holds, and the equation f (x) = x has two equal real roots. Find the analytic expression of F (x) Let f (1-x) = f (1 + x) hold,
- 20. Given the function f (x) = x 2 + ax + 3, when x ∈ [- 2,2], f (x) ≥ A is constant, and the minimum value of a is obtained