1. Tardigrade
  2. Question
  3. Mathematics
  4. A coplanar beam of light emerging from a point source have the equation λ x - y +2(1+λ)=0, ∀ λ ∈ R; the rays of the beam strike an elliptical surface and get reflected inside the ellipse. The reflected rays form another convergent beam having the equation μ x - y +2(1-μ)=0, ∀ μ ∈ R. Further it is found that the foot of the perpendicular from the point (2,2) upon any tangent to the ellipse lies on the circle x2+y2-4 y-5=0. The area of the largest triangle that an incident ray and corresponding reflected ray can enclose with major axis of the ellipse is equal to

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. A coplanar beam of light emerging from a point source have the equation $\lambda x - y +2(1+\lambda)=0, \forall \lambda \in R$; the rays of the beam strike an elliptical surface and get reflected inside the ellipse. The reflected rays form another convergent beam having the equation $\mu x - y +2(1-\mu)=0, \forall \mu \in R$. Further it is found that the foot of the perpendicular from the point $(2,2)$ upon any tangent to the ellipse lies on the circle $x^2+y^2-4 y-5=0$.
The area of the largest triangle that an incident ray and corresponding reflected ray can enclose with major axis of the ellipse is equal to

Conic Sections

Solution:

Area of $\Delta PF _1 F _2=\frac{1}{2} \times$ base $\times$ height $=\frac{1}{2} \times 4 \times$ height maximum are $=$ maximum height $= b$
$\because b^2=a^2\left(1-e^2\right)=9\left(1-\frac{4}{9}\right) $
$\Rightarrow b=\sqrt{5} $
$\Rightarrow \text { Area }=2 \sqrt{5}$

Solution Image