Introduction:
We are given two parabolas with equations y^2 = 4ax and y^2 = 4c(x – b). We need to prove that these two parabolas cannot have a common normal, other than the axis unless b/(a – c) > 2.

Assumption:
Let's assume that the two parabolas have a common normal other than the axis.

Equations of Tangents:
The equation of the tangent to the parabola y^2 = 4ax at the point (at^2, 2at) is given by y = mx + a/t, where m is the slope of the tangent.
Similarly, the equation of the tangent to the parabola y^2 = 4c(x – b) at the point (ct^2 + b, 2ct) is given by y = mx + c/t.

Condition for Common Normal:
For the two parabolas to have a common normal, the slopes of the tangents at the points of intersection must be equal.

Points of Intersection:
To find the points of intersection, we equate the y-coordinates of the two parabolas:
2at = 2ct
=> a = c

Condition for Equal Slopes:
Now, let's equate the slopes of the tangents:
m1 = a/t
m2 = c/t

For the slopes to be equal, we have:
a/t = c/t
=> a = c

Condition for Common Normal:
Since we assumed a = c, the two parabolas can only have a common normal if a = c.

Condition for b/(a – c) > 2:
If a = c, then b/(a – c) is undefined. Therefore, for the two parabolas to have a common normal other than the axis, we must have b/(a – c) > 2.

Conclusion:
In conclusion, the two parabolas y^2 = 4ax and y^2 = 4c(x – b) cannot have a common normal, other than the axis, unless b/(a – c) > 2. This is because for the two parabolas to have a common normal, we must have a = c, and in that case, b/(a – c) is undefined.

Hi unblock me .its me