Question
The chord $$AB$$ of the parabola $${y^2} = 4ax$$ cuts the axis of the parabola at $$C.$$ If $$A = \left( {at_1^2,\,2a{t_1}} \right),\,B = \left( {at_2^2,\,2a{t_2}} \right)$$ and $$AC:AB = 1:3$$ then :
A.
$${t_2} = 2{t_1}$$
B.
$${t_2} + 2{t_1} = 0$$
C.
$${t_1} + 2{t_2} = 0$$
D.
none of these
Answer :
$${t_2} + 2{t_1} = 0$$
Solution :
$$\eqalign{ & {\text{Here, }}C = \left( {\frac{{2at_1^2 + at_2^2}}{3},\,\frac{{4a{t_1} + 2a{t_2}}}{3}} \right) \cr & {\text{It line on }}y = 0 \cr & \therefore \,\,\frac{{4a{t_1} + 2a{t_2}}}{3} = 0 \cr} $$
$$\eqalign{ & {\text{Here, }}C = \left( {\frac{{2at_1^2 + at_2^2}}{3},\,\frac{{4a{t_1} + 2a{t_2}}}{3}} \right) \cr & {\text{It line on }}y = 0 \cr & \therefore \,\,\frac{{4a{t_1} + 2a{t_2}}}{3} = 0 \cr} $$