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)

$2{{t}_{1}}{{t}_{2}}={{t}_{1}}+{{t}_{2}}$

Answer 2

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 ${{t}_{1}}={{t}_{2}}$