Derivation of Standard and Explicit Coordinate Equations for Axis Aligned Parabolas

The following gives the derivation for Standard Coordinate Equation for Parabolas having their Directrix Parallel to \(X\)-Axis.
Let's consider a Parabola having it's Focus at a point \((x_f,y_f)\), Vertex at \((x_v,y_v)\) and having Directrix given by the equation \(y=C_D\) or \(y-C_D=0\) where \(C_D\) is the constant of equation. Now,

Distance of any point (\(x,y\)) on the Parabola from the Focus = \(\sqrt{{(x-x_f)}^2 + {(y-y_f)}^2}\)

Also, Distance of any point (\(x,y\)) on the Parabola from the Directrix = \(y-C_D\)

Now, as per the definition of Parabola, any point (\(x,y\)) on the Parabola must be equidistant from the Directrix and the Focus. Therefore,

\(\sqrt{{(x-x_f)}^2 + {(y-y_f)}^2} = y-C_D\)

\(\Rightarrow {(x-x_f)}^2 + {(y-y_f)}^2 = {(y-C_D)}^2\)

\(\Rightarrow {(x-x_f)}^2 = {(y-C_D)}^2 - {(y-y_f)}^2\)

\(\Rightarrow {(x-x_f)}^2 = y^2 - 2C_Dy + {C_D}^2 - y^2 - {y_f}^2 + 2y_fy\)

\(\Rightarrow {(x-x_f)}^2 = - 2C_Dy + {C_D}^2 - {y_f}^2 + 2y_fy\)   ...(1)

\(\Rightarrow {(x-x_f)}^2 = 2y(y_f-C_D) + (C_D-y_f) (C_D+y_f) \)

\(\Rightarrow {(x-x_f)}^2 = 2y(y_f-C_D) - (y_f-C_D) (y_f+C_D) \)   ...(2)

Now, the quantity \(y_f-C_D\) is the Signed Distance between the Focus and the Directrix, which is 2 times the Signed Focal Length \(f\) of the Parabola. That is,

\(y_f-C_D=2f\)   ...(3)

Since this Parabola has Directrix Parallel to \(X\)-Axis, the \(X\)-Coordinate of the Vertex of this Parabola is same as the \(X\)-Coordinate of the Focus. Hence,

\(x_v=x_f\)   ...(4)

Also, since the Vertex lies in the Mid Point of the Focus and Directrix, the \(Y\)-Coordinate of Vertex is given as

\(\frac{(y_f+C_D)}{2}=y_v\)

\(y_f+C_D=2y_v\)   ...(5)

Now, using equations (2), (3), (4) and (5) we get

\({(x-x_v)}^2 = 2y(2f) - (2f)(2y_v) \)

\(\Rightarrow {(x-x_v)}^2 = 4fy - 4fy_v \)

\(\Rightarrow {(x-x_v)}^2 = 4f(y - y_v)\)   ...(6)

The equation (6) given above is the Standard Coordinate Equation for Parabolas having their Directrix Parallel to \(X\)-Axis. If the Vertex is at Origin (0,0), this equation becomes

\( x^2 = 4fy\)   ...(7)

If the value of the Signed Focal Length \(f > 0\) then the Parabola Opens Upwards in Positive Direction of Y-Axis.
If the value of the Signed Focal Length \(f < 0\) then the Parabola Opens Downwards in Negative Direction of Y-Axis.
The following gives the derivation for Explicit Coordinate Equation for Parabolas having their Directrix Parallel to \(X\)-Axis.

From equation (1) given above we have

\({(x-x_f)}^2 = - 2C_Dy + {C_D}^2 - {y_f}^2 + 2y_fy\)

Expanding the LHS of the above equation and rearranging we get

\(x^2 -2x_fx - 2(y_f - C_D)y + {x_f}^2 + {y_f}^2 - {C_D}^2 = 0\)

Now, as given in equation (3) above, the quantity \(y_f-C_D\) is the Signed Distance between the Focus and the Directrix, which is 2 times the Signed Focal Length \(f\) of the Parabola. Therefore the above equation becomes,

\(x^2 -2x_fx - 2(2f)y + {x_f}^2 + {y_f}^2 - {C_D}^2 = 0\)

\(\Rightarrow x^2 -2x_fx - 4fy + {x_f}^2 + {y_f}^2 - {C_D}^2 = 0\)

\(\Rightarrow 4fy=x^2 -2x_fx + {x_f}^2 + {y_f}^2 - {C_D}^2\)

\(\Rightarrow y=\frac{1}{4f}x^2 -\frac{2x_f}{4f}x + \frac{({x_f}^2 + {y_f}^2 - {C_D}^2)}{4f}\) ...(8)

Now Let Constants,

\(A=\frac{1}{4f}\)

\(B=-\frac{2x_f}{4f}\)

\(C=\frac{({x_f}^2 + {y_f}^2 - {C_D}^2)}{4f}\)

Substituting constants \(A\),\(B\) and \(C\) in equation (8) we get

\(y=Ax^2 + Bx + C\) ...(9)

The equation (9) given above is the Explicit Coordinate Equation for Parabolas having their Directrix Parallel to \(X\)-Axis.

If the value of Coefficient of \(x^2\hspace{.3cm} A > 0\) (i.e when the value of Signed Focal Length \(f > 0\)) in equation (9) then the Parabola Opens Upwards in Positive Direction of Y-Axis.
If the value of Coefficient of \(x^2\hspace{.3cm} A < 0\) (i.e when the value of Signed Focal Length \(f < 0\)) in equation (9) then the Parabola Opens Downwards in Negative Direction of Y-Axis.
The following gives the derivation for Standard Coordinate Equation for Parabolas having their Directrix Parallel to \(Y\)-Axis.
Let's consider a Parabola having it's Focus at a point \((x_f,y_f)\), Vertex at \((x_v,y_v)\) and having Directrix given by the equation \(x=C_D\) or \(x-C_D=0\) where \(C_D\) is the constant of equation. Now,

Distance of any point (\(x,y\)) on the Parabola from the Focus = \(\sqrt{{(x-x_f)}^2 + {(y-y_f)}^2}\)

Also, Distance of any point (\(x,y\)) on the Parabola from the Directrix = \(x-C_D\)

Now, as per the definition of Parabola, any point (\(x,y\)) on the Parabola must be equidistant from the Directrix and the Focus. Therefore,

\(\sqrt{{(x-x_f)}^2 + {(y-y_f)}^2} = x-C_D\)

\(\Rightarrow {(y-y_f)}^2 + {(y-y_f)}^2 = {(x-C_D)}^2\)

\(\Rightarrow {(y-y_f)}^2 = {(x-C_D)}^2 - {(x-x_f)}^2\)

\(\Rightarrow {(y-y_f)}^2 = x^2 - 2C_Dx + {C_D}^2 - x^2 - {x_f}^2 + 2x_fx\)

\(\Rightarrow {(y-y_f)}^2 = - 2C_Dx + {C_D}^2 - {x_f}^2 + 2x_fx\)   ...(10)

\(\Rightarrow {(y-y_f)}^2 = 2x(x_f-C_D) + (C_D-x_f) (C_D+x_f) \)

\(\Rightarrow {(y-y_f)}^2 = 2y(x_f-C_D) - (x_f-C_D) (x_f+C_D) \)   ...(11)

Now, the quantity \(x_f-C_D\) is the Signed Distance between the Focus and the Directrix, which is 2 times the Signed Focal Length \(f\) of the Parabola. That is,

\(x_f-C_D=2f\)   ...(12)

Since this Parabola has Directrix Parallel to \(Y\)-Axis, the \(Y\)-Coordinate of the Vertex of this Parabola is same as the \(Y\)-Coordinate of the Focus. Hence,

\(y_v=y_f\)   ...(13)

Also, since the Vertex lies in the Mid Point of the Focus and Directrix, the \(X\)-Coordinate of Vertex is given as

\(\frac{(x_f+C_D)}{2}=x_v\)

\(x_f+C_D=2x_v\)   ...(14)

Now, using equations (11), (12), (13) and (14) we get

\({(y-y_v)}^2 = 2x(2f) - (2f)(2x_v) \)

\(\Rightarrow {(y-y_v)}^2 = 4fx - 4fx_v \)

\(\Rightarrow {(y-y_v)}^2 = 4f(x - x_v)\)   ...(15)

The equation (15) given above is the Standard Coordinate Equation for Parabolas having their Directrix Parallel to \(Y\)-Axis. If the Vertex is at Origin (0,0), this equation becomes

\(y^2 = 4fx\)   ...(16)

If the value of the Signed Focal Length \(f > 0\) then the Parabola Opens Rightwards in Positive Direction of X-Axis.
If the value of the Signed Focal Length \(f < 0\) then the Parabola Opens Leftwards in Negative Direction of X-Axis.
The following gives the derivation for Explicit Coordinate Equation for Parabolas having their Directrix Parallel to \(Y\)-Axis.

From equation (10) given above we have

\({(y-y_f)}^2 = - 2C_Dx + {C_D}^2 - {x_f}^2 + 2x_fx\)

Expanding the LHS of the above equation and rearranging we get

\(y^2 -2y_fy - 2(x_f - C_D)x + {x_f}^2 + {y_f}^2 - {C_D}^2 = 0\)

Now, as given in equation (12) above, the quantity \(x_f-C_D\) is the Signed Distance between the Focus and the Directrix, which is 2 times the Signed Focal Length \(f\) of the Parabola. Therefore the above equation becomes,

\(y^2 -2y_fy - 2(2f)x + {x_f}^2 + {y_f}^2 - {C_D}^2 = 0\)

\(\Rightarrow y^2 -2y_fy - 4fx + {x_f}^2 + {y_f}^2 - {C_D}^2 = 0\)

\(\Rightarrow 4fx =y^2 -2y_fy + {x_f}^2 + {y_f}^2 - {C_D}^2\)

\(\Rightarrow x=\frac{1}{4f}y^2 -\frac{2y_f}{4f}y + \frac{({x_f}^2 + {y_f}^2 - {C_D}^2)}{4f}\) ...(17)

Now Let Constants,

\(A=\frac{1}{4f}\)

\(B=-\frac{2y_f}{4f}\)

\(C=\frac{({x_f}^2 + {y_f}^2 - {C_D}^2)}{4f}\)

Substituting constants \(A\),\(B\) and \(C\) in equation (17) we get

\(x=Ay^2 + By + C\) ...(18)

The equation (18) given above is the Explicit Coordinate Equation for Parabolas having their Directrix Parallel to \(Y\)-Axis.

If the value of Coefficient of \(y^2\hspace{.3cm} A > 0\) (i.e when the value of Signed Focal Length \(f > 0\)) in equation (18) then the Parabola Opens Rightwards in Positive Direction of X-Axis.
If the value of Coefficient of \(y^2\hspace{.3cm} A < 0\) (i.e when the value of Signed Focal Length \(f < 0\)) in equation (18) then the Parabola Opens Leftwards in Negative Direction of X-Axis.

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