Let the normal to parabola y2=4ax at P meets the curve again in Q . If PQ and the  normal at Q makes angles α and β respectively with the positive x -axis in positive  direction, then tan⁡α(tan⁡α+tan⁡β) is equal to

Let P=at12,2at1&Q=at22,2at2 Equation of the normal at Pt1 is y+xt1=2at1+at13………….(1) Slope of the normal at Pt1 is −t1=tan⁡α Similarly Slope of the normal at Qt2 is −t2=tan⁡β (1) is passing through Qat22,2at2⇒2at2+at22t1=2at1+at13⇒2at2−t1=at1t12−t22⇒−2at1−t2=at1t1+t2t1−t2⇒−2=t1t1+t2⇒−2=−tan⁡α(−tan⁡α−tan⁡β)⇒tan⁡α(tan⁡α+tan⁡β)=−2