Suppose F is a fixed point in a plane and suppose L is a fixed line in the plane with F not on L. Now suppose P is the locus of a point Q in the plane such that Q is the center of a circle C passing through the point F and tangent to the line L.
●Explain why P is a parabola.
●Assuming that F and L lie in an xy-plane with F=(0,1) and L the x-axis, show that P is a translation of a radial scaling of the parabola P*:x^2-y=0.
有冇人可以幫我解答下-MATH
2008-10-15 9:00 pm
回答 (2)
2008-10-15 10:38 pm
✔ 最佳答案
Let the fixed line L be the x -axis for simplicity sake, in fact it can be generalized to any line by translation and rotation of the x-axis.Let the point Q be (x,y) and the fixed point F be (a,b). Since F is on the circle with Q as centre and L is tangential to the circle, so distance between QF and the distance from Q to the line is the same.
That is (x-a)^2 + (y-b)^2 = y^2
(x-a)^2 = y^2 - (y-b)^2 = y^2 - y^2 + 2by - b^2 = 2by - b^2
y= [(x - a)^2 + b^2]/2b = (x-a)^2/2b + b/2 which is a quadratic function, , P
is a parabola.
Now a= 0, b = 1,
so y = (x^2 + 1)/2
y = x^2/2 + 1/2
which is a translation of 1/2 unit upwards and scaling of 1/2 of y = x^2.
參考: my maths knowledge
2008-10-15 9:37 pm
Let the fixed line L be the x -axis for simplicity sake, in fact it can be generalized to any line by translation and rotation of the x-axis. Let the point Q be (x,y) and the fixed point F be (a,b). Since F is on the circle with Q as centre and L is tangential to the circle, so distance between QF and the distance from Q to the line is the same. That is (x-a)^2 + (y-b)^2 = y^2
(x-a)^2 = y^2 - (y-b)^2 = y^2 - y^2 + 2by - b^2 = 2by - b^2
y= [(x - a)^2 + b^2]/2b = (x-a)^2/2b + b/2 which is a quadratic function, therefore, P is a parabola.
Now a= 0, b = 1,
so y = (x^2 + 1)/2
y = x^2/2 + 1/2
which is a translation of 1/2 unit upwards and scaling of 1/2 of y = x^2.
(x-a)^2 = y^2 - (y-b)^2 = y^2 - y^2 + 2by - b^2 = 2by - b^2
y= [(x - a)^2 + b^2]/2b = (x-a)^2/2b + b/2 which is a quadratic function, therefore, P is a parabola.
Now a= 0, b = 1,
so y = (x^2 + 1)/2
y = x^2/2 + 1/2
which is a translation of 1/2 unit upwards and scaling of 1/2 of y = x^2.
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