A) \[ST=SG.SP\]
B) \[ST\ne SG=SP\]
C) \[ST=SG\ne SP\]
D) \[ST=SG=SP\]
Correct Answer: D
Solution :
Let a point\[P(a{{t}^{2}},2at)\]on the parabola \[{{y}^{2}}=4ax\] on which the equations of tangent and normal are\[ty=x+a{{t}^{2}}\] and \[y=-tx+2at+a{{t}^{3}}\]respectively. Since, tangent and normal meet the\[x-\]axis at T and G respectively. Therefore, coordinates of T and G are\[(-a{{t}^{2}},0)\]and\[(2a+a{{t}^{2}},0)\]respectively. By the definition of the parabola, \[SP=PM=a+a{{t}^{2}}\] \[SG=VG-VS\] \[=2a+a{{t}^{2}}-a=a+a{{t}^{2}}\] and \[ST=VS+VT=a+a{{t}^{2}}\] Thus, \[SP=SG=ST\]You need to login to perform this action.
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