JEE Main & Advanced Mathematics Conic Sections Equation of the Chord Joining any Two Points on the Parabola 

Let \[P(at_{1}^{2},2a{{t}_{1}}),Q(at_{2,}^{2},2a{{t}_{2}})\]be any two points on the parabola \[{{y}^{2}}=4ax\]. Then, the equation of the chord joining these points is, \[y-2a{{t}_{1}}=\frac{2}{{{t}_{1}}+{{t}_{2}}}\left( x-at_{1}^{2} \right)\] or \[y({{t}_{1}}+{{t}_{2}})=2x+2a{{t}_{1}}{{t}_{2}}\].

(1) Condition for the chord joining points having parameters \[{{\mathbf{t}}_{\mathbf{1}}}\] and \[{{\mathbf{t}}_{\mathbf{2}}}\] to be a focal chord: If the chord joining points \[(at_{1}^{2},2a{{t}_{1}})\] and \[(at_{2}^{2},2a{{t}_{2}})\] on the parabola passes through its focus, then \[(a,0)\] satisfies the equation \[y({{t}_{1}}+{{t}_{2}})=2x+2a{{t}_{1}}{{t}_{2}}\]

\[\Rightarrow \] \[0=2a+2a{{t}_{1}}{{t}_{2}}\] \[\Rightarrow \] \[{{t}_{1}}{{t}_{2}}=-1\] or \[{{t}_{2}}=-\frac{1}{{{t}_{1}}}\].

(2) Length of the focal chord: The length of a focal chord having parameters \[{{t}_{1}}\]and \[{{t}_{2}}\]for its end points is \[a{{({{t}_{2}}-{{t}_{1}})}^{2}}\].