Find the condition for a line other than y-axis to exist such that it is tangent to $y^2=4ax$ and normal to $x^2=4by$.


For tangent line: $$x-y{t_1}+a{t_1}^2=0$$ where $t_1$ is parameter. For normal line: $$yt_2 +x-{t_2}(2b+b{t_2}^2)=0$$ where $t_2$ is another parameter.

Now for them to be same if $$ {\frac{t_2}{-t_1}}={\frac{{t_2}({-2b-b{t_2}^2})}{a{t_1}^2}}=1$$

So we get after simplifying

$$b{t_1}^2+a{t_1}+2b=0$$ My doubt is that-

What I want to ask is that for the real solutions of the equation we will take discriminant to be greater than 0. But will the discriminant equal to zero will hold the tangent and normal property (as given in the question)? What is the physical significance of discriminant equal to zero?


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