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Given the hyperbola x ^ 2-4y ^ 2 = 4 and the point m (8,1), the straight line passing through the electric m intersects the hyperbola at two points a and B, and M is the midpoint of the line AB, the equation of the straight line is obtained
Let a (a, b), B (C, d)
a^2 - 4b^2 = 4 (1)
c^2 - 4d^2 = 4 (2)
M (8,1) is the midpoint of line AB, 8 = (a + C) / 2,1 = (B + D) / 2
a+c = 16,b+d = 2
(1)-(2):(a+c)(a-c) = 4(b+d)(b-d)
16(a-c) = 4*2(b-d)
(b-d)/(a-c) = 2
That is, the slope of AB is 2, and the equation is Y - 1 = 2 (x - 8)
y = 2x - 15
The line passing through point P (8,1) intersects hyperbola x2-4y2 = 4 at two points a and B, and P is the midpoint of line ab. the equation of line AB is obtained
Let a (x1, Y1), B (X2, Y2), then X1 + x2 = 16, Y1 + y2 = 2, ∵ x12-4y12 = 4, x22-4y22 = 4, ∵ 16 (x1-x2) - 8 (y1-y2) = 0, ∵ KAB = 2, ∵ the equation of straight line is Y-1 = 2 (X-8), that is, 2x-y-15 = 0
The line passing through point P (8,1) intersects hyperbola x2-4y2 = 4 at two points a and B, and P is the midpoint of line ab. the equation of line AB is obtained
Let a (x1, Y1), B (X2, Y2), then X1 + x2 = 16, Y1 + y2 = 2, ∵ x12-4y12 = 4, x22-4y22 = 4, ∵ 16 (x1-x2) - 8 (y1-y2) = 0, ∵ KAB = 2, ∵ the equation of straight line is Y-1 = 2 (X-8), that is, 2x-y-15 = 0
Through point a (6,1), make a straight line L and hyperbola 16 / x square - 4 / y square = 1, intersect at BC, and a and the midpoint of line BC, find l equation (point difference method)
Let B (x1, Y1), C (X2, Y2) and a (6,1) be the midpoint of BC, then: X1 + x2 = 12, Y1 + y2 = 2
Substituting B and C into hyperbola, we get the following result:
x1²/16-y1²/4=1
x2²/16-y2²/4=1
Results: (x1 & # 178; - x2 & # 178;) / 16 - (Y1 & # 178; - Y2 & # 178;) / 4 = 0
Then: (Y1 & # 178; - Y2 & # 178;) / (x1 & # 178; - x2 & # 178;) = 1 / 4
That is: (y1-y2) (Y1 + Y2) / (x1-x2) (x1 + x2) = 1 / 4
That is: 2 (y1-y2) / 12 (x1-x2) = 1 / 4
So: (y1-y2) / (x1-x2) = 3 / 2
That is: K (BC) = 3 / 2
Another point a (6,1)
So the equation of L is: 3x-2y-16 = 0
The test shows that there are two intersections between the straight line and hyperbola
So the equation of L is: 3x-2y-16 = 0
RELATED INFORMATIONS
- 1. If the line and hyperbola x2-4y2 = 4 intersect at two points a and B, if the midpoint coordinate of line AB is (8,1), then the equation of the line is______ ．
- 2. Given the line l1:3x + 4Y = 6 and l2:3x-4y = - 6, then the relationship between the inclination angles of line L1 and L2 is
- 3. Given the straight line L1: 3x-4y-17 = 0, L2: 3x-4y + 23 = 0, find the equation of circle C tangent to L1, L2 and X axes
- 4. When the line L passes through the point P (2, - 3), the inclination angle is 45 ° larger than that of the line y = 2x-1. The equation of line L is obtained
- 5. It is known that a straight line m and a straight line L: y = 2x + 3 have the same intercept on the y-axis, and the inclination angle is twice that of L. The equation for finding a straight line m is given
- 6. If a straight line is drawn through the point P (1,2) and its inclination angle is twice that of the straight line L: x-y-3 = 0, then the equation of the straight line is______ ．
- 7. The line L passing through the point P (1,2,) divides the circle x2 + y2-4x-5 = 0 into two arches. When the area of the smaller arch is the smallest, the equation of the line L is______ ．
- 8. Given that the straight line L passes through the point (2,1), the inclination angle is twice of the inclination angle of the straight line x-radical 3Y + 3 = 0, the equation for solving the straight line L is obtained
- 9. It is known that circle C: x ^ 2 + y ^ 2-2x + 4y-4 = 0. Is there any symmetry between two points a and B on circle C about the straight line y = kx-1, and the circle with diameter AB passing through the origin?
- 10. Given that x ^ 2 + y ^ 2 = 1 and the line y = 2x + m intersect a and B, and the angles between OA, OB and the positive direction of X axis are a and B respectively, it is proved that sin (a + b) is a fixed value
- 11. The line passing through point P (8,1) intersects hyperbola x2-4y2 = 4 at two points a and B, and P is the midpoint of line ab. the equation of line AB is obtained
- 12. A. B is two points on hyperbola x ^ 2-y ^ 2 / 2 = 1, and point (1,2) is a line segment ab. the midpoint of the line AB equation is obtained
- 13. It is known that m (4,2) is the midpoint of line AB cut by ellipse x2 + 4y2 = 36, then the equation of line L is______ ．
- 14. If the midpoint of line a and B is (1,1), then the equation of L is?
- 15. The linear equation with ellipse (x ^ 2 / 9) + (y ^ 2 / 4) = 1 intersecting at two points a and B, and the midpoint of line AB is m (1,1) is, using the point difference method
- 16. Let the equation of the ellipse be x square + y square / 4 = 1, the straight line passing through M (0,1) intersects the ellipse at two points AB, O is the coordinate origin, Op vector = 1 / 2 (OA vector + ob vector). When l rotates around point m, the trajectory equation of the moving point P is obtained
- 17. If the parabola C1 and the parabola C2: y ^ 2 = - 4x are symmetric with respect to the straight line x + y = 2, then the focal coordinate of the parabola C1 is
- 18. Given circle C1: (x + 3) ^ 2 + y ^ 2 = 1 and circle C2: (x-3) ^ 2 + y ^ 2 = 9, the moving circle m is circumscribed with circle C1 and C2 at the same time The orbit equation of moving circle center M
- 19. Given that two circles C1: (x + 4) 2 + y2 = 2, C2: (x-4) 2 + y2 = 2, the moving circle m is tangent to both circles C1 and C2, then the trajectory equation of the moving circle center m is () A. X = 0b. X22-y214 = 1 (x ≥ 2) C. x22-y214 = 1D. X22-y214 = 1 or x = 0
- 20. It is known that the line L passes through the point P (- 2,5), and the slope is - 3 / 4. The equation of the circle where the line L is tangent to the point (2,2) and the center of the circle is on the line x + Y-11 = 0 is obtained Who knows? Reply