If the points a- 0-a t12- 2 a t1 and a t22- 2 a t2 are collinear- write the value of t1 t2-

For the points (a, 0), (at12, 2at1) and (at22, 2at2) to be collinear, the following condition has to be met: a01at122at11at222at21=0 #8658;a2at1 2at2 0+12a2t ...

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Byju's Answer

Standard XII

Mathematics

Area of Triangle with Coordinates of Vertices Given

If the points...

Question

If the points (a, 0), (at₁², 2at₁) and (at₂², 2at₂) are collinear, write the value of t₁ t₂.

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Solution

For the points (a, 0), (at₁², 2at₁) and (at₂², 2at₂) to be collinear, the following condition has to be met:

$|\begin{matrix} a & 0 & 1 \\ a {t_{1}}^{2} & 2 a t_{1} & 1 \\ a {t_{2}}^{2} & 2 a t_{2} & 1 \end{matrix}| = 0 \Rightarrow a (2 a t_{1} - 2 a t_{2}) - 0 + 1 (2 a^{2} {t_{1}}^{2} t_{2} - 2 a^{2} t_{1} {t_{2}}^{2}) = 0 \Rightarrow 2 a^{2} (t_{1} - t_{2}) + 2 a^{2} t_{1} t_{2} (t_{1} - t_{2}) = 0 \Rightarrow 2 a^{2} (t_{1} - t_{2}) (1 + t_{1} t_{2}) = 0$

$\Rightarrow (t_{1} - t_{2}) = 0 or (1 + t_{1} t_{2}) = 0 (a \neq 0) \Rightarrow 1 + t_{1} t_{2} = 0 (∵ t_{1} \neq t_{2}) \Rightarrow t_{1} t_{2} = - 1$

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If the points (a, 0), (at12, 2at1) and (at22, 2at2) are collinear, write the value of t1 t2.

For the points (a, 0), (at12, 2at1) and (at22, 2at2) to be collinear, the following condition has to be met: a01at122at11at222at21=0⇒a2at1-2at2-0+12a2t12t2-2a2t1t22=0⇒2a2t1-t2+2a2t1t2t1-t2=0⇒2a2t1-t21+t1t2=0 ⇒t1-t2=0 or 1+t1t2=0 a≠0⇒1+t1t2=0 ∵t1≠t2⇒t1t2=-1

If the points (a, 0), (at₁², 2at₁) and (at₂², 2at₂) are collinear, write the value of t₁ t₂.