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Brief Review of Cartesian System of Rectangular Coordinates — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSBrief Review of Cartesian System of Rectangular Coordinates1 Mark
Question
If the point $(\text{a},0)(\text{a}\text{t}_1^2, 2\text{a}\text{t}_2)\text{ and }(\text{a}\text{t}_2^2, 2\text{a}\text{t}_2)$ are collinear, write the value of $\text{t}_1\text{t}_2.$
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Answer

Let three given point be$\text{A}=(\text{x}_1,\text{y}_1)=(\text{a},0)$
$\text{B}=(\text{x}_2,\text{y}_2)=(\text{a}\text{t}_1^2, 2\text{a}\text{t}_1)$
$\text{ and }\text{C}=(\text{x}_3,\text{y}_3)=(\text{a}\text{t}_2^2, 2\text{a}\text{t}_2)$
If the given points are collinear, then
$\text{x}_1(\text{y}_2-\text{y}_3)+\text{x}_2(\text{y}_3-\text{y}_1)+\text{x}_3(\text{y}_1-\text{y}_2)=0$
$\Rightarrow\text{a}(2\text{a}\text{t}_1-2\text{a}\text{t}_2)+\text{a}\text{t}_1^2(2\text{a}\text{t}_1-0)+\text{a}\text{t}_2^2(0-2\text{a}\text{t}_1)=0$
$\Rightarrow 2\text{a}^2\text{t}^1-2\text{a}^2\text{t}^2+2\text{a}^2\text{t}_1^2\text{t}_2-2\text{a}^2\text{t}_1^2\text{t}_1=0$
$\Rightarrow 2\text{a}^2\text{t}_1(1+\text{t}_1\text{t}_2)-2\text{a}^2$
im co.

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