Correct sequence in development is :

The given curve is ay² = x³
The point of tangency is (at², at³)
Let the point that tangent intersects again be (as², as³)
We can find the slope of tangent by differentiating the given curve w.r.t x


Now, we will use two point form to solve the question.
If (x₁, y₁) and (x₂, y₂) be two points on the line then the slope is

For this question, it will be

We will solve it further and find the value.


(s - t) = 0
s = t
It means the same point of tangency. We want to find different point.
(2s + t) = 0
s = 
So, the point will be

This is the required answer.

The curves are as follows:
ax² + by² = 1            ...(1)
Ax² + By² = 1          ...(2)        
We will take the derivative of equation (1) w.r.t x.
2ax + 2by=0
2by= -2ax

This is this slope of first curve. We will call it slope1.
In the same way we will take the derivative of equation (2) w.r.t x 
2Ax + 2By = 1

We will call it slope2.
The two curves are orthogonal. So, the product of their slope is -1.
Slope1 × Slope2 = -1

We will substitute (3) in (1)

We will substitute (4) in (2)

Divide (5) by (6) 

This is the required answer.
 
 
 
.