Parabola y = - x ^ 2 + (m-2) x + 3 (m-1) passing through point (3,0) Find the vertex coordinates of this parabola?

Parabola y = - x ^ 2 + (m-2) x + 3 (m-1) passing through point (3,0) Find the vertex coordinates of this parabola?

Substituting point (3, m) into the analytical formula, we can get: - 3 ^ 2 + (m-2) * 3 + 3 (m-1) = 0
-9+3m-6+3m-3 =0
m =3
So the analytical formula of parabola is: y = - x ^ 2 + X + 6 = - (x ^ 2-x-6) = - (x ^ 2-x + 1 / 4) + 25 / 4 = - (x-1 / 2) ^ 2 + 25 / 4
So the vertex coordinates of parabola: (1 / 2,25 / 4)

The straight line with the slope of - 1 passes through the focus of the parabola y square = 4x, intersects with the parabola at two points a and B, and calculates the length of line ab

The focus of the parabola y square = 4x is (1,0). Let the line be y = - x + B. let the point (1,0) be B = 1, so the line is y = - x + 1, y square = 4x (- x + 1) square = 4x x square - 6x + 1 = 0 (x1-x2) square = (x1 + x2) square - 4x1x2 = 36-4 = 32. Similarly, (y1-y2) square = 32, so the length of AB = radical [(x1-x2

What is the trajectory equation of the midpoint of a straight line with a slope of 2 cut by a parabola x ^ 2 = y?

y=2x+b
Substituting, then x2 = 2x + B
x2-2x-b=0
x1+x2=2
y=2x+b
So Y1 + y2 = 2x1 + B + 2x2 + B = 2 (x1 + x2) + 2B = 4 + 2B
For the middle point, x = (x1 + x2) / 2 = 1
y=(y1+y2)/2=2+b
The equation x2-2x-b = 0 has two solutions
So 1 + b > 0
b>-1
So x = 1, Y > 1

If the two metaphysics of a circle are parallel to each other, then the arcs between the two strings are equal

It is known that AB, CD are two strings in circle O, and AB / / CD
Verification: arc AC = arc BD
Proof: Link Ad
Because AB / / CD,
So angle a = angle D,
So arc AC = arc BD (in the same circle, the arcs opposite by equal circular angles are equal)

If the two strings of a circle are parallel to each other, are the arcs between the two strings equal? Why

Make a diameter parallel to the two strings through the center of the circle, and use the property of parallel line to get the result

If the arcs between the two strings are equal, are the two strings of a circle parallel? Why?

Not necessarily. There are too many possibilities. You can draw them at will. For example, if you draw two diameters, they are also strings. The length of the arc between them is semicircle, but they intersect

In a circle, if the arcs between two parallel lines are equal, then whether the arcs between a tangent line and a chord are equal

It's equal, of course

If the arcs between the two strings of a circle are equal, are the two strings parallel to each other true or false? A line perpendicular to the string (note the line)
The line perpendicular to the string (note the line) bisects the string and bisects the two arcs it faces
Why are these two sentences wrong,

It's all wrong
1. For example, the arcs of two diameters are equal, but they are not parallel
2. The chord should be bisected perpendicular to the diameter of the chord, and the two arcs of the chord should be bisected

How to prove equal arc equal chord

The center angles of equal arcs are equal
Because it is an equal arc, the radius is equal
The two radii are equal, and the center angle is equal (if it is a superior arc, the 360 center angle is equal)
Two triangles are congruent

It is proved that the diameter of the bisector string is perpendicular to the string and the arc of the bisector string is opposite

Let AB be the chord on the circle O; let CD be the diameter on O and bisect AB at point E; let the bisector OE on the bottom of the isosceles triangle OAB be the height on the bottom; let CD be perpendicular to AB; angle AOD = angle BOD; so arc ad = arc BD