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Curves – Parabola
Definition: Given a fixed point and fixed line, a parabola is a collection of all the points that the distance to the fixed point is equal to the distance to the fixed line.
The point is known as focal point; the line is known as directrix.
Then, we consider to formulate the equation for parabola.
the focus: F(0,a) the directrix L: y+d = 0
the distance to F is equal to the distance to the line L: x2 + (y-a)2 = (y+d)2 x2 + (-2ay) + a2 = 2ady + d2 y = ( x2 + (a2 – d2) ) y= x2 +
So, for the parabola with the form y = Ax2 + Bx + C, we shall be able to find its focal point and directrix easily according to the derivation above.
Theorem: For the parabola y = Ax2 + Bx + C, its focal point and directrix are as follows:
the focal point is ( , ) ; the directrix is y + = 0 .
Proof:
y = Ax2 + Bx + C = A(x2 + x + ( )2 ) + C - = A(x + )2 + C - = A(x + )2 +
use the previous derivation, we have a + d = a – d =
a = d =
The focal point is ( , ) ; The directrix is y + = 0 .
And the parabola is symmetric to the line x = . We call it the axis of the parabola.
Theorem ( Tangent Line Equation ): Let A(x0, y0) be a point on the parabola y= Ax2 + Bx + C . The equation of the tangent line crossing the parabola at A is
y – y0 = ( B+2Ax0 )( x – x0 )
Proof:
The tangent line is crossing the parabola at A. Thus, it has the form
y-y0 = m ( x-x0 )
( for this type of parabolas, it does not have tangent line of the form x = k )
Furthermore, the tangent line crosses the parabola only at 1 point, i.e. , the point A.
y-y0 = m ( x-x0 ) y= Ax2 + Bx + C
Hence, the system equation above only has 1 solution ( repeated solutions ).
m(x-x0) + y0 = Ax2 + Bx + C Ax2 + ( B-m ) x + ( C + mx0 –y0 ) = 0 has repeated roots . (B-m)2 – 4A( C+mx0-y0 ) = 0
Do not forget (x0, y0) on the parabola; that means
y0 = Ax02 + Bx0 + C
Thus, m2 – (2B+4Ax0)m + (B+2Ax0)2 = 0 m = (B+2Ax0) , (B+2Ax0)
Hence, the tangent line equation is
y – y0 = ( B+2Ax0 )( x – x0 )
Short cut:
Using the result from Calculus, the slope of tangent line can be found via the derivative function: y= Ax2 + Bx + C = 2Ax + By
slope for the tangent line at (x0, y0) is (2Ax0+By0)
Theorem ( Reflection ) : Consider the parabola y=Ax2 + Bx + C and a point P(a,b) on the parabola. Let F be its focal point. If L is a tangent line crossing the parabola at P, and the line T crossing P is parallel to the axis of the parabola as shown below, then
FPQ = SPR
Proof:
The tangent line is crossing the parabola at P.
Then the normal vector of the tangent line is
= ( 2Aa+B, -1 )
Furthermore, we have
Focal point : F ( , ) T : x=a
= ( , ) b=Aa2 + Ba + C
= ( , ) = ( , )
And consider = ( , ) the vector parallel to line T : = ( 0, )
Please notice that is always aligned with the direction of the “mouth” of the parabola.
We check the inner products of those unit vectors : ( the inner product of two unit vectors is equal the cosine of the angle between the two vectors)
= =
They are of the same cosine value.
For the parabola, it is sufficient to say FPQ = SPR
This theorem has its meaning on physics. If you emit the light from the focal point of the mirror with the shape parabola, the reflective light will be parallel to the axis of the parabola.
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