0
$\begingroup$

Question

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

Attempt

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?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.