This is a result which will save a lot of time if you can remember it. We will derive it in this solution.
Any line with slope m can be written as y = mx + c. ...(i)
We want to find the value of C.
The equation of normal at (acosθ,bsinθ) is
axcosθ−bysinθ=a2−b2
We can rearrange it as
Comparing (1) and (2) we get
abtanθ=m and −(a2−b2)sinθb=c⇒To find c,we have to find sinθTanθ=bma
⇒sinθ=bm√a2+b2m2⇒c=−(a2−b2)bxbm√a2+b2m2⇒y=mx−m(a2−b2)√a2+b2m2 is the equation of normal.