JEE Main & Advanced Mathematics Conic Sections Point of Intersection of Tangents at Any Two Points on The Parabola

(1) The point of intersection of tangents at two points \[P(at_{1}^{2},2a{{t}_{1}})\] and \[Q(at_{2}^{2},2a{{t}_{2}})\] on the parabola \[{{y}^{2}}=4ax\] is \[(a{{t}_{1}}{{t}_{2}},a({{t}_{1}}+{{t}_{2}}))\].

(2) The locus of the point of intersection of tangents to the parabola \[{{y}^{2}}=4ax\] which meet at an angle \[\alpha \] is \[{{(x+a)}^{2}}{{\tan }^{2}}\alpha ={{y}^{2}}-4ax\].

(3) Director circle: The locus of the point of intersection of perpendicular tangents to a conic is known as its director circle. The director circle of a parabola is its directrix.

(4) The tangents to the parabola \[\frac{x{{x}_{1}}}{{{a}^{2}}}-\frac{y{{y}_{1}}}{{{b}^{2}}}=1\] at \[P(at_{1}^{2},2a{{t}_{1}})\] and \[Q(at_{2}^{2},2a{{t}_{2}})\] intersect at R. Then the area of triangle \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is \[\frac{1}{2}{{a}^{2}}{{({{t}_{1}}-{{t}_{2}})}^{3}}\].