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Two diagonals of rectangle ABCD intersect at M (2,0) AB side, the linear equation is x-3y-6 = 0 point t (- 1,1) on the straight line of AD, the moving circle P passes through n Two diagonals of rectangle ABCD intersect at the edge of M (2,0) AB, and the linear equation is x-3y-6 = 0 point t (- 1,1). On the straight line where ad is located, the moving circle P passes through n (- 2,0) and is circumscribed with the circumscribed circle of ABCD, and the trajectory equation of the center of the moving circle is obtained
Because the moving circle P passes through the point n, so | PN | is the radius of the circle, and because the moving circle P is the outer circle of the circle m, so | PM | = | PN | + 2, that is | PM | - | PN | = 2; therefore, the trajectory of point P is a circle with m, n as the focus and the real axis length of 2. Because the real half axis length a = root sign 2, the half focal length C = 2. The imaginary half axis length b = root sign 2
It is known that the center of the square is m (1,4) and a vertex is a (0,2)
The linear equation of Ma is y = 2x + 2 and the slope is 2
Then the equation of the line passing through M and perpendicular to Ma is
Y = - 0.5x + 4.5 (the other two intersections of the square are on this line)
Make a circle with m point as the center and Ma as the radius
（X-1）^2+(Y-4)^=(4-2)^2+(1-0)^2=5
Intersection with y = - 0.5x + 4.5
Two intersections are the other two vertices of the square
B(3,3)C(-1,5)
The AB equation is y = x / 3 + 2
The AC equation is y = - 3x + 2
Make a straight line through the point m (0,1) so that the line segment cut by two straight lines L1: x-3y + 10 = 0, L2: 2x + Y-8 = 0 is exactly bisected by m, and solve the linear equation
Let B (T, 8-2t) and m (0, 1) be the midpoint of AB, and a (- t, 2t-6) can be obtained from the midpoint coordinate formula
Make a straight line through the point m (0,1) so that the line segment cut by two straight lines L1: x-3y + 10 = 0, L2: 2x + Y-8 = 0 is exactly bisected by m, and solve the linear equation
Let the obtained line and the known lines L1 and L2 intersect at two points a and B respectively. ∵ point B is on the line l2:2x + Y-8 = 0, so let B (T, 8-2t). And m (0, 1) is the midpoint of AB, and a (- t, 2t-6) is obtained from the midpoint coordinate formula. ∵ point a is on the line L1: x-3y + 10 = 0, ∵ (- t) - 3 (2t-6) + 10 = 0, and T = 4. ∵ B (4, 0), a (- 4, 2), so the obtained linear equation is: x + 4y-4 =0．
RELATED INFORMATIONS
- 1. Make a straight line through the point m (0,1) so that the line segment cut by two straight lines L1: x-3y + 10 = 0, L2: 2x + Y-8 = 0 is exactly bisected by m, and solve the linear equation
- 2. Given that two points a and B are on the line 2x-y = 0 and X + 2Y = 0 respectively, and the midpoint of AB segment is p (0,5), then the length of AB segment is______ ．
- 3. Make a straight line L through point a (- 1,1) so that the midpoint of the line segment cut by two parallel lines L1: x + 2y-1 = 0 and: x + 2y-3 = 3 is exactly on the straight line L3: x-y-1 = 0, The equation for finding the line L
- 4. The center is the origin, and the two focal points have a point P (3, t) on the ellipse on the x-axis. If the distance between the point P and the two focal points is 6.5 and 3.5 respectively, the elliptic equation is solved
- 5. If the center of the moving circle is on the parabola y2 = 8x and the moving circle is always tangent to the straight line x + 2 = 0, then the moving circle must pass the fixed point------ Another problem: in the plane rectangular coordinate system xoy, two fixed points a and B on the parabola y2 = 4x which are different from the coordinate origin o are satisfied with AO vertical Bo, and the trajectory equation of the center of gravity g of the triangle AOB is obtained How to answer these two questions? How to find out?
- 6. The center F of the circle M: x = 1 + cos θ is a parabola e: x = 2pt & sup2; the focal point of the circle M: x = 1 + cos θ is a straight line intersection parabola e passing through the focal point F at two points y = sin θ y = 2pt Find the value range of afbf The center F of the circle M: x = 1 + cos θ y = sin θ is a parabola e: x = 2pt & sup2; the focal point of y = 2pt passes through the straight line intersection parabola e of the focal point F, and the value range of AF · BF is calculated at two points ab
- 7. Given circle O: X & # 178; + Y & # 178; = 1 and point m (4,2) The equation for finding the straight line L through the tangent line L of the circle O in the direction of point m The equation for a circle m with chord length 4, which is cut by a straight line y = 2x-1 and centered on M Let p be any point in the middle circle m, passing through point P and leading tangent to circle O, and the tangent point is Q. try to explore whether there is a certain point R in the plane, so that PQ / PR is a fixed value? If there is, give an example. If not, explain the reason
- 8. Find the following trajectory equation (1) for the center of circle m and circle C: (x + 2) &# 178; + Y & # 178; = 2, which is inscribed and passes through point a (2,0)
- 9. Find the center coordinate of the circle: X & # 178; + Y & # 178; - 2x + 4y-11 = 0, and the radius is
- 10. If x + 5Y = 6, then x2 + 5xy + 30y=______ ．
- 11. It is known that the equation of the line where the edge CD of square ABCD is located is x-3y-4 = 0, the intersection point of diagonal AC and BD is p (5.2) It is known that the equation of the line where the edge CD of square ABCD is located is x-3y-4 = 0, the intersection point of diagonal AC and BD is p (5.2) Solving the equation of the line where AB edge is located The equation for finding the circumcircle of square ABCD
- 12. The distance from the focus of the parabola y = 4x2 to the collimator is______ ．
- 13. If the line passing through the focus F of the parabola y2 = 4x intersects with the parabola at two points a and B, then OA · ob=______ ．
- 14. The parabola y = - 3 / 4x ^ 2 + 3 intersects the x-axis at points a and B, the straight line y = - 3 / 4x + B intersects at points B and C, and the straight line y = - 3 / 4x + B intersects the y-axis at points E (1) Write the analytical expression of the straight line BC (2) Finding the area of △ ABC (3) If point m moves from a to B at the speed of one unit length per second on line AB (not coincident with a and b), and point n moves from B to C at the speed of two units length per second on ray BC. Let the motion time be T seconds, write out the functional relationship between the area s of △ MNB and T, and find out the maximum area of △ MNB when point m moves for how long?
- 15. The image of quadratic function y = x ^ 2-2x 5 is folded along the y-axis to obtain the analytical expression of the parabola
- 16. The parabola y = ax square + BX + C, when x = 2, x = 0, the value of Y is equal The parabola y = ax square + BX + C, when x = 2, x = 0, the value of Y is equal, the vertex m of the parabola is on the straight line y = 3x-7, and the abscissa of the intersection point with the straight line is 4 Find the analytic formula of this parabola
- 17. Given the parabola y = ax square + BX + C, | a | = 1 / 2, the coordinates of the highest point are (* 1,5 / 2), find the value of a.b.c
- 18. If the parabola y = ax square + BX + C passes through (2,0) (0,2) (- 1,0) 3 points, its analytical formula is
- 19. It is known that the square of parabola y = ax + BX + C (a is less than 0) passes through point a (- 2,0), O (0,0) We know the square of parabola y = ax + BX + C (a)
- 20. If the distance between a point m on the parabola y ^ 2 = 3x and the coordinate origin o is 2, then the distance between the point m and the focus of the parabola is 2_____ .