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Condition P: the sum of the distances from the moving point m to two fixed points is equal to the fixed length, condition Q: the trajectory of the moving point m is an ellipse, and condition P is the result of condition Q A. Necessary and sufficient condition B. necessary and insufficient condition C. sufficient and unnecessary condition D. neither sufficient nor necessary condition
b
Given the ellipse X & # 178 / 4 + Y & # 178 / 25 = 1, find the linear equation of the chord with (1,4) as the midpoint
Ellipse: 25X & # 178; + 4Y & # 178; = 100
Let the chord be AB, a (x1, Y1) B (X2, Y2)
x1+x2=2,y1+y2=8
that
25x1²+4y1²=100
25x2²+4y2²=100
Subtraction of two formulas
25(x1²-x2²)+4(y1²-y2²)=0
25(x1+x2)(x1-x2)+4(y1-y2)(y1+y2)=0
25*2+4*8*(y1-y2)/(x1-x2)=0
(y1-y2)/(x1-x2)=(y-4)/(x-1)
For a long time
25+16×(y-4)/(x-1)=0
25x-25+16y-64=0
25X + 16y-89 = 0
Through the ellipse X & # - 178 / 16 + Y & # - 178 / 4 = 1, find the linear equation of (1) the chord with P (2, - 1) as the midpoint
(2) The trajectory equation of the midpoint of a parallel chord with slope 2 (3) the trajectory equation of the midpoint of a chord cut by an ellipse through a straight line Q (8,2) focuses on the latter two questions
X ^ 2 / 16 + y ^ 2 / 4 = 1 (1), string abxa + XB = 2xp = 4, Ya + Yb = - 2K (AB) = (Ya Yb) / (XA XB) [(XA) ^ 2 / 16 + (ya) ^ 2 / 4] - [(XB) ^ 2 / 16 + (Yb) ^ 2 / 4] = 1-1 = 0 (XA + XB) * (XA XB) + 4 (Ya + Yb) * (Ya Yb) = 0 (XA + XB) + 4 (Ya + Yb) * (Ya Yb) / (XA XB) = 0k (AB) = (Ya Yb) / (XA XB) =
The linear equation of the chord with a point m (1,1) in the ellipse x216 + y24 = 1 as the midpoint is ()
A. 4x-3y-3=0B. x-4y+3=0C. 4x+y-5=0D. x+4y-5=0
Let the linear equation be Y-1 = K (x-1), and substitute it into the ellipse x216 + y24 = 1. By simplifying, we can get x216 + (KX − K + 1) 24 = 1, (4k2 + 1) x2 + 8 (k-k2 & nbsp;) x + 4k2-8k-12. Therefore, the linear equation is & nbsp; Y-1 = - 14 (x-1), that is, x + 4y-5 = 0, so D
Through a point m (2.1) in the ellipse X & # 178; / 16 + Y & # 178; / 4 = 1, a string is introduced, so that the string is bisected by point m, and the equation of the string is obtained?
Use the point difference method
Let the ends of chord be a (x1, Y1), B (X2, Y2),
Then X1 ^ 2 / 16 + Y1 ^ 2 / 4 = 1, X2 ^ 2 / 16 + Y2 ^ 2 / 4 = 1,
Then (x2-x1) (x2 + x1) / 16 + (y2-y1) (Y2 + Y1) / 4 = 0,
Because m is the midpoint of AB, X1 + x2 = 4, Y1 + y2 = 2,
Substituting (x2-x1) / 4 + (y2-y1) / 2 = 0,
The solution is k = (y2-y1) / (x2-x1) = - 1 / 2,
So the equation is Y-1 = - 1 / 2 * (X-2),
It is reduced to x + 2y-4 = 0
If there is a point P (2,1) in the ellipse e: x216 + y24 = 1, then the linear equation of the chord passing through P and taking P as the midpoint is___ ．
Let the straight line intersect the ellipse at a (x1, Y1), B (X2, Y2), then x1216 + y124 = 1, x2216 + y224 = 1. By subtracting the two formulas, we get (x1 + x2) (x1-x2) 16 + (Y1 + Y2) (y1-y2) 4 = 0. Then X1 + x2 = 4, Y1 + y2 = 2, ∧ KAB = y1-y2x1-x2 = - 12. Therefore, the linear equation is Y-1 = - 12 (X-2), that is, x + 2y-4 = 0
What is the chord length of a straight line X-Y + 1 = 0 cut by an ellipse X & # 178 / 16 + Y & # 178 / 4 = 1?
X-Y + 1 = 0y-y + 1 = 0y = 1 = 0y = x + 1 = 0y + 1 = 0y + 1 = 0y + 1 = 0 y = 1 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\178; + 412
The chord length of the line L passing through P (- 1,0) with an inclination angle of π / 3 and the ellipse X & # 178; + 2Y & # 178; = 4 is
The equation of solving the line L with P (- 1,0) inclination angle π / 3 is y-0 = √ 3 (x + 1). Let the chord length of the line L and the ellipse X & # 178; + 2Y & # 178; = 4 be AB, and a (x1, Y1), B (X2, Y2), then AB = √ (x1-x2) &# 178; + (y1-y2) &# 178; = √ (1 + K & # 178;) * / x1-x2 / = √ (1 + K & # 178;) *
What is the chord length of an ellipse X & # 178; + Y & # 178; = 1 cut by a straight line y = X-1
Radical 2
Find the equation of the straight line of the chord AB with the point m (2, - 1) in the ellipse 5x ^ 2 + 8y ^ 2 = 40 as the midpoint
It's a process
Let's talk about the train of thought
Let's set the coordinates of AB, let's say (x1, Y1) (X2, Y2)
Then the two points are on the ellipse and we get two equations. If we make a difference, we get equation I
Then we get X1 + x2 = 4, Y1 + y2 = - 2 from the key coordinates and substitute them into I,
The straight line equation passing through AB point can be obtained
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