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In the plane rectangular coordinate system, the parabola y = AX2 + BX + C and X axis intersect at point a, B two points, a is on the left side of B, ab = 3, intersects with y axis at point C, and OC = 2ao, OC = ob, then what is the value of B? Our teacher said there are four answers, good reward 100
If the origin o is on the left side of a, then the known | OC | = 6,
So a (3,0), B (6,0), C (0,6) or C (0, - 6),
Therefore, let y = a (x-3) (X-6) and substitute the c-coordinate, we can get a = ± 1 / 3,
So y = ± (x ^ 2-9x + 18) / 3, B = ± 3;
If the origin o is on the right side of B, this is impossible;
If the origin o is between a and B, then a (- 1,0), B (2,0), C (0,2) or C (0, - 2),
So we can substitute x = 1 with x = 1,
Therefore, y = ± (x ^ 2-x-2), B = ± 1
To sum up, the value of B is - 1 / 3, or - 1, or 1 / 3, or 1
As shown in the figure, point C is a point on the line L = x in the first quadrant, and OC = radical 2, the line y = 2x + 1 intersects the y-axis with the point a, intersects the x-axis at the point B, and the line AB is along the x-axis Direction translation and through point C, find the analytical formula of the line after translation
If point C is a point on the line L = x in the first quadrant, and OC = radical 2, then C (1,1) is a point on the line L = x in the first quadrant,
From the line y = 2x + 1, let the line after translation be y = 2x + B, and substitute C (1,1) into b = - 1,
So the analytic formula of the line after translation is y = 2x-1
As shown in Fig
(1) Y = - 2x + 4, substituting y = 0 to get x = 2, a (2,0) substituting x = 0 to get y = 4, C (0,4)
(2) Let D (2, y), according to the properties of folding, we can get CD = ad = y, BD = 4-y, 2 2 + (4)
-y) 2 = y 2, y = 2.5
Let the analytic formula of the straight line CD be y = KX + 4, substituting x = 2, y = 2.5, we can get k = - 0.75 the analytic formula of straight line CD is y = - 0.75x + 4
(3) Point O meets the requirements, P1 (0,0)
② The symmetry point of point o with respect to AC also meets the requirements of point P, there are ∠ ACP = ∠ BAC = ∠ ACO, P can be on the straight line CD, and let P (x, - 0.75x + 4), (X-2) 2 + (- 0.75x + 4) 2] = 2? And the solution x = 3.2 ﹥ P2 (3.2,1.6)
③ The symmetry point B about AC is also a P-point that meets the requirements of the P point, PQ
As shown in the figure, the known point C is a point in the first quadrant of the straight line y = X. the line y = 2x + 1 intersects the Y axis at point a and the intersection X axis at point B. the straight line AB is along the ray OC Two units of root sign are translated along the direction of ray OC, and the analytical formula of straight line after translation is obtained
y=2x
As shown in Fig. 1, it is known that the line y = - 2x + 4, X axis and Y axis intersect at points a and C respectively, and take OA and OC as the edges, and make them in the first quadrant It is known that the line y = - 2x + 4 intersects points a and C with X axis and Y axis respectively, and makes a rectangular oabc in the first quadrant with OA and OC as the edges (1) Find the coordinates of points a and C; (2) Fold △ ABC in half so that the point a coincides with the point C, and the crease intersects AB at the point d to find the analytic formula of the straight line CD (Fig. 2); (4) Whether there is a point P (except point B) in the coordinate plane so that △ APC and △ ABC are congruent. If so, ask for the coordinates of all points P that meet the conditions. If not, please explain the reasons I don't understand question (4)
(1) When y = 0, the coordinate of 0 = - 2x + 4 ν A is (2,0)
When x = 0, x = 4
(2) Let ad = x ν BD = 4 - x CD? 2 = 4 + 16 + X? - 8x
∵CD=AD ∴4+16+x²－8x=x² ∴x=2.5
Let the analytic formula of CD be y = KX + B (K and B are constants, K ≠ 0)
According to the meaning of the title, 4 = B ﹥ k = - 0.75
2.5=2k+b b=4
The analytic formula of the function is y = - 0.75x + 4
(3) There were P1 (0,0), P2 (16 / 5,8 / 5), P3 (- 6 / 5,12 / 5)
Make P2 ⊥ X axis, extend CB to F, and let P2 coordinate as (x, y)
From the meaning of the title, (X-2) 2 + y 2 = 2
(4－y)²+x²=4²
∴x=2y
Substituting y = - 0.75x + 4 gives y = - 0.75 × 2Y + 4
ν y = 8 / 5, x = 16 / 5 ℅ the coordinates of P2 are (16 / 5, 8 / 5)
Similarly, the coordinates of P3 are (- 6 / 5,12 / 5)
As shown in Fig (1) Find the coordinates of points a and C; (2) fold △ ABC so that the point a coincides with point C, and the crease intersects AB at point d to find the analytic formula of the straight line CD (Fig. 2); (3) in the coordinate plane, is there a point P (except point B) so that △ APC and △ ABC are congruent? If so, ask for the coordinates of all points P that meet the conditions; if not, please explain the reasons
A(4,0);C(0,8)
y=0.75x+8
Origin o, (- 2.4,4.8)
As shown in the figure, in the plane rectangular coordinate system xoy, the straight line AB and X axis intersect at point a, and Y axis intersect at point B, and OA = 3, ab = 5 As shown in the figure, in the plane rectangular coordinate system xoy, the straight line AB and X axis intersect at point a and Y axis intersect at point B, and OA = 3, ab = 5. Point P starts from point O and moves along OA at the speed of 1 unit per second to point a, and immediately returns along Ao at the original speed after reaching point a; point Q starts from point a and moves along AB at a speed of 1 unit per second to point B at a uniform speed, When point Q reaches point B, the point P will stop. The time for point P and Q to move is T seconds (T > 0) (1) Find the analytic formula of line ab; (2) In the process of point P moving from O to a, the functional relationship between the area s and t of △ Apq is obtained (it is not necessary to write out the range of T); (3) In the process of point e moving from B to o, the following problems are completed: ① Can the quadrilateral qbed become a right angle trapezoid? If so, ask for the value of T; if not, please explain the reason; ② When de passes through point O, write the value of t directly There must be a process in the second item of sub question (3). I mainly look at this question. If there is no point, no point will be given
Finally, the idea of the last question: first, construct isosceles from the middle vertical line = convert to equal time = equal distance structure isosceles = three lines in one, out of the midpoint = median line = conversion to the midpoint
∵ ed ⊥ PQ and DP = DQ ᙽ OPQ is an isosceles triangle ∵ OP = AQ ᚉ OQ = aq
The △ OQA is to wait for △ to be the midpoint F of OA and connect FQ ∵ ⊥ OQ = AQ ᚉ FQ ⊥ Ao in OQA
ν q is the midpoint of AB, t = 5 / 2
As shown in the figure, in the plane rectangular coordinate system xoy, the line AB intersects with the axis at point a and axis Y intersects with point B, and OA = 3, ab = 5, Point P starts from point O and moves along OA at the speed of 1 unit per second to point A. after arriving at point a, it immediately returns to point Q at the original speed along Ao. Starting from point a, point AB moves at a uniform speed to point B at a speed of 1 unit per second. With the movement of P Q, de keeps vertical bisection of PQ, and PQ intersects with point D. the broken line qb-bo-op starts from point E and stops moving when point B reaches point Q Point P also stops, and the movement time of point p q is T seconds (T > 0) (1) Finding the analytic formula of line ab (2) In the process of point P moving from O to a, the area of the ball △ Apq, the functional relationship between S and t (it is not necessary to write out the value range of T) (3) In the process of point e moving from B to o, the following problems are completed: ① Can quadrilateral qebd become right angle trapezoid? If yes, request the value of T. if not, please explain the reason ② When de passes through point O, please write the value of t directly!
(1) In RT △ AOB, OA = 3, ab = 5, OB = ab2-oa2 = 4. A (3,0), B (0,4) is obtained by Pythagorean theorem. Let the analytic formula of straight line AB be y = KX + B. {3K + B = 0b = 4. The analytic formula of line AB is y = - 43b = 4. The analytic formula of line AB is y = - 43x + 4; (2) as shown in Fig. 1, QF ⊥ Ao is made at point f. ∵ AQ = OP = t
In the plane rectangular coordinate system, the straight line AB intersects with x-axis and y-axis at a (3,0) B (0, root sign 3) respectively. C is the segment AB, and the moving point passes through C to make CD, and the x-axis is perpendicular to d If the trapezoid obcd is equal to 4 times the root sign 3 / 3, find the point C coordinate. 3. Whether there is a point P in the 1 quadrant so that P, O, B are the vertices of the triangle similar to the triangle AOB? If yes, ask for all the coordinates of P that meet the conditions. If not, please explain the reason!
The second question is: how can trapezoid be equal to a number? The third problem is as long as the hypothesis exists, and then compare the corresponding sides of two triangles to get all the coordinates of point P (because there are common edges OB and points o and B are fixed), There are three such points in total, one coincides with a, one does not meet the requirements in the fourth quadrant, and the other is another point corresponding to the right triangle with point B as the perpendicular foot. Question: area of trapezoid answer: the second question, assuming that the coordinates of point C are (M, n), the area formula has (n + root 3) times m times 1 / 2 = 4 times root 3, Another equation, two equations, two unknowns, OK
Adopt it
As shown in the figure, in the plane rectangular coordinate system, ⊙ P and X axis intersect at two points a and B respectively, and the coordinates of point P are (3, - 1), ab =2 3． (1) Find the radius of ⊙ P (2) Translate ⊙ P downward and find the translation distance when ⊙ P is tangent to X axis
(1) Connect PA and make PC ⊥ AB at point C. according to the vertical diameter theorem, the following results are obtained
AC=1
2AB=1
2×2
3=
Three
In the right angle △ PAC, it is obtained from Pythagorean theorem that pa2 = PC2 + ac2
PA2=12+（
3）2=4
∴PA=2
The radius of P is 2;
(2) The distance between the point P and the X axis is equal to the radius
The translation distance is: 2-1 = 1
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