Several standard locus results involving conics appear repeatedly in JEE. These form the "ready reckoner" of conic-related loci.

Locus of the foot of perpendicular from focus to tangent: For a parabola $y^2$=4ax, this locus is the tangent at the vertex (x=0, the y-axis). For an ellipse or hyperbola $x^2$/$a^{2+}$/-$y^2$/$b^2$=1, the locus is the auxiliary circle x^{2+y}^2=$a^2$.

Locus of intersection of perpendicular tangents $\frac{director circle}{directrix}$: For a circle x^{2+y}^2=$r^2$, it is x^{2+y}^2=2$r^2$. For an ellipse $x^2$/a^{2+y}^2/$b^2$=1, it is x^{2+y}^2=a^{2+b}^2. For a hyperbola $x^2$/a^{2-y}^2/$b^2$=1, it is x^{2+y}^2=a^{2-b}^2 (exists only when a>b). For a parabola $y^2$=4ax, it is the directrix x=-a.

Locus of the midpoint of a focal chord: For the parabola $y^2$=4ax, the locus is $y^2$=2a(x-a).

Locus of the intersection of tangents at the ends of a focal chord: For $y^2$=4ax, the tangents at ($at1^2$,2at1) and ($at2^2$,2at2) with t1*t2=-1 meet at (-a, a(t1+t2)). The locus is x=-a, the directrix.

The T=S1 method for chord midpoint problems: for a chord of a circle x^{2+y}^2=$r^2$ with midpoint (h,k), the chord equation is hx+ky=h^{2+k}^2. Apply the additional condition (passes through a point, subtends a given angle, etc.) to obtain the locus.

Apollonius circle: If PA/PB=k (constant, k!=1) where A and B are fixed points, the locus is a circle. When k=1, the locus degenerates to the perpendicular bisector of AB.