Question

The three distinct points $ A(at_{1}^{2},\,2a{{t}_{1}}),\,\,B(at_{2}^{2},\,\,2a{{t}_{2}}) $ and $ C(0,\,a) $ (where $a$ is a real number) are collinear, if

• A. $ {{t}_{1}}{{t}_{2}}=-1 $
• B. $ {{t}_{1}}{{t}_{2}}=1 $
• C. $ 2{{t}_{1}}{{t}_{2}}={{t}_{1}}+{{t}_{2}} $
• D. $ {{t}_{1}}+{{t}_{2}}=a $

Answer 1

Answer: (C)

If there points $ A(at_{1}^{2},2a{{t}_{1}}),\,\,\,B(at_{2}^{2},\,2a{{t}_{2}}) $ and $ C(0,a), $ collinear, if
$ \left| \begin{matrix} at_{1}^{2} & 2a{{t}_{1}} & 1 \\ at_{2}^{2} & 2a{{t}_{2}} & 1 \\ 0 & a & 1 \\ \end{matrix} \right|=0 $
Use operation; $ {{R}_{2}}\to {{R}_{2}}-{{R}_{1}},{{R}_{3}}\to {{R}_{3}}-{{R}_{1}} $ $ \left| \begin{matrix} at_{1}^{2} & 2a{{t}_{1}} & 1 \\ a(t_{2}^{2}-t_{1}^{2}) & 2a({{t}_{2}}-{{t}_{1}}) & 0 \\ -at_{1}^{2} & a-2a{{t}_{1}} & 0 \\ \end{matrix} \right|=0 $
Expand with respect to $ {{C}_{3}} $
$ a({{t}_{2}}-{{t}_{1}})\,({{t}_{2}}+{{t}_{1}})\,(a-2a{{t}_{1}}) $
$ +2{{a}^{2}}t_{1}^{2}({{t}_{2}}-{{t}_{1}})=0 $
$ \Rightarrow $ $ a({{t}_{2}}-{{t}_{1}})\{({{t}_{1}}+{{t}_{2}})\,(a-2a{{t}_{1}})+2at_{1}^{2}\}=0 $
$ \Rightarrow $ $ a({{t}_{2}}-{{t}_{1}})\,\{a{{t}_{1}}+a{{t}_{2}}-2at_{1}^{2} $
$ -2a{{t}_{1}}{{t}_{2}}+2at_{1}^{2}\}=0 $
$ \Rightarrow $ $ a({{t}_{2}}-{{t}_{1}})\,(a{{t}_{1}}+a{{t}_{2}}-2a{{t}_{1}}{{t}_{2}})=0 $
$ \Rightarrow $ $ {{a}^{2}}({{t}_{2}}-{{t}_{1}})\,({{t}_{1}}+{{t}_{2}}-2{{t}_{1}}+{{t}_{2}})=0 $
$ \Rightarrow $ $ {{t}_{2}}-{{t}_{1}}=0 $
or $ {{t}_{1}}+{{t}_{2}}-2{{t}_{1}}{{t}_{2}}=0 $
$ \Rightarrow $ $ {{t}_{1}}={{t}_{2}} $
or $ {{t}_{1}}={{t}_{2}} $