the conditions for the parabola y 2 =4(x-d) & y 2 =4ax have a common normal other than x axis (a0,c0) is ac0)?

Question

the conditions for the parabola y 2 =4(x-d) & y 2 =4ax have a common normal other than x axis (a>0,c>0) is a>c>0)?

Riddhish Bhalodia · Answer

we have many equations
let the common normal intersect P1 at (x1,y1) and P2 at (x2,y2)

1) we have
$y_1^2=4(x_1-d)$
2) we have
$y_2^2=4ax_2$
3) By equating the nomal slope at points, and the slope of the line
we get
$-y_2/2ax_2 = -y_1/2x_1 = (x_2-x_1)/(y_2-y_1)$

eliminate any three variables and we get a quadratic in the remainig variable, it’s D>=0
that will give you the condition

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the conditions for the parabola y 2 =4(x-d) & y 2 =4ax have a common normal other than x axis (a>0,c>0) is a>c>0)?

the conditions for the parabola y²=4(x-d) & y²=4ax have a common normal other than x axis (a>0,c>0) is a>c>0)?

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