Hard Practice — Advanced Set

Quick Recap — Multi-concept Coordinate Geometry (JEE Main level)

Real JEE problems chain several ideas. Combine locus formation, tangency and chord-of-contact ($T=0$, $SS_1=T^2$), focal-chord relations ($t_1t_2=-1$), director circles, and reflection/normal properties. Many are numerical-value (Section B) — compute an exact number.

• Perpendicular tangents meet on the director circle $x^2+y^2=a^2+b^2$ (ellipse) or $x^2+y^2=2a^2$ (circle radius $a$).
• Length of chord of contact from an external point $=\dfrac{2Lr}{\sqrt{L^2+r^2}}$, with tangent length $L=\sqrt{S_1}$.
• Normal to $y^2=4ax$ at $t_1$ meets the curve again at $t_2=-t_1-\dfrac{2}{t_1}$.