If t1,t2 and t3 distinct. and the points (t1.2at1+at13).(t2.2at2+at23),(t3.2at3+at33) are collinear, then t1+t2+t3=

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If t1,​t2​ and t3​ distinct. and the points (t1​.2at1​+at1​3).(t2​.2at2​+at2​3),(t3​.2at3​+at3​3) are collinear, then t1​+t2​+t3​=

Filo · Accepted Answer

Let points are P(t1​.2at1​+at13​) ,Q(t1​,2at2​+at13​) and R(t3​,2at3​+at33​)P,Q and R are collinear⇒area ( ΔPQR) =0⇒ ∣∣​ 111​t1​t2​t2​​2at1​+at13​ 2at2​+at23​2at3​+at33​​∣∣​=0⇒ ∣∣​111​t1​t2​t3​​2at1​2at2​2at3​​∣∣​+∣∣​111​t1​t2​t3​​at13​at23​at33​ ​∣∣​=0⇒2a∣∣​111​t1​t2​t3​​t1​t2​t3​ ​∣∣​+a∣∣​111​t1​t2​t3​​t13​t23​t33​​∣∣​=0Determinant is zero if two rows of a determinant are same ⇒0+a ∣∣​111​t1​t2​t3​​t13​ t23​t33​ ​∣∣​=0R2​→R2​−R1​         R3​→R3​−R1​⇒ ∣∣​1 10​t1​t2​−t1​t2​−t1​​t13​t23​−t13​t33​−t13​ ​∣∣​=0⇒(t2​−t1​)(t3​−t1​) ∣∣​100​t1​11​t13​t12​+t1​t2​+t22​t12​+t1​t3​+t32​​∣∣​=0⇒(t2​−t1​)(t2​−t1​)[t12​+t!​t2​+t32​−t12​−t1​t2​−t22​]=0⇒(t2​−t1​)(t3​−t1​)[t23​+t1​t3​−t1​t2​−t22​]=0⇒(t2​−t1​)(t3​−t1​)(t3​−t2​)(t1​+t2​+t3​)=0as t1​= t2​ = t3​⇒t2​−t1​ or t3​−t1​ or t3​−t2​ can't be zero⇒t1​+t2​+t3​=0

Question Text	If $t_{1,} t_{2}$ and $t_{3}$ distinct. and the points $(t_{1} .2 a t_{1} + a t_{1}^{3}) . (t_{2} .2 a t_{2} + a t_{2}^{3}), (t_{3} .2 a t_{3} + a t_{3}^{3})$ are collinear, then $t_{1} + t_{2} + t_{3} =$
Answer Type	Text solution:1
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If $t_{1,} t_{2}$ and $t_{3}$ distinct. and the points $(t_{1} .2 a t_{1} + a t_{1}^{3}) . (t_{2} .2 a t_{2} + a t_{2}^{3}), (t_{3} .2 a t_{3} + a t_{3}^{3})$ are collinear, then $t_{1} + t_{2} + t_{3} =$

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If t1,​t2​ and t3​ distinct. and the points (t1​.2at1​+at1​3).(t2​.2at2​+at2​3),(t3​.2at3​+at3​3) are collinear, then t1​+t2​+t3​=

Text solutionVerified

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If $t_{1,} t_{2}$ and $t_{3}$ distinct. and the points $(t_{1} .2 a t_{1} + a t_{1}^{3}) . (t_{2} .2 a t_{2} + a t_{2}^{3}), (t_{3} .2 a t_{3} + a t_{3}^{3})$ are collinear, then $t_{1} + t_{2} + t_{3} =$