A hyperbola's transverse axis and a parabolas axis are same will they meet

Question

\begin{align*} y &= A x ^ { 2} \\ ( y - 2) ^ { 2} - x ^ { 2} &= 1 \end{align*} I understand the intersection in lower branch but not in the upper branch. What if we alter the value of A , will it still happen? Also please tell me if there is a method to know that two parabolas with same axis will intersect or not. Thank you

Answer 1

Let us assume that $A$ is positive.

Solve

$$( y - 2) ^ { 2} - x ^ { 2}= 1$$ for $ y$ to get $$ y= 2\pm\sqrt {x^2+1}$$

The hyperbola and the parabola meet at points were $$Ax^2 = 2\pm\sqrt {x^2+1}$$ Note that at $x=0$ the right hand side is positive and the left side is $0$

For large values of $ x$ the left side, $Ax^2$, will dominate and we are going to have 4 points of intersections.

Regarding your question about intersection of parabolas, note that it all depends on the equations of your parabolas.

If you have two parabolas, $y=Ax^2$ and $y=Bx^2$ with positive $A$ and $B$, the only point of intersection is the common vertex $(0,0)$.