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Coordinate Geometry — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsCoordinate Geometry1 Mark
Question
State whether the following statements are true or false. Justify your answer.
Points A(-6, 10), B(-4, 6) and C(3, -8) are collinear such that
$\text{AB}=\frac{2}{9}=\text{AC}.$
✓

Answer

True:Points A, B and C will be collinear if $\text{ar}(\triangle\text{ABC})=0$
$\text{ar}\triangle\text{ABC}=\frac{1}{2}[\text{x}_1(\text{y}_2-\text{y}_3)+\text{x}_2(\text{y}_3-\text{y}_1)+\text{x}_3(\text{y}_1-\text{y}_2)]$
$\Rightarrow\frac{1}{2}[-6\{6-(-8)\}-4(-8-10)+3(10-6)]=0$
$\Rightarrow-6(14)-4(-18)+3(4)=0$
$\Rightarrow-84+72+12=0$
$\Rightarrow-84+84=0,$ which is true
So, points A, B and C are collinear.
$\text{AC}^2=(\text{x}_2-\text{x}_1)^2+(\text{y}_2-\text{y}_1)^2$
$\Rightarrow\text{AC}^2=(3+6)^2+(-8-10)^2$
$\Rightarrow\text{AC}^2=81+324$
$\Rightarrow\text{AC}=\sqrt{405}$
$\Rightarrow\text{AC}=9\sqrt{5}\text{ units}$
$\Rightarrow\text{AB}^2=[-4-(-6)]^2+(6-10)^2$
$\Rightarrow\text{AB}^2=(-4+6)^2+(-4)^2$
$\Rightarrow\text{AB}^2=4+16$
$\Rightarrow\text{AB}^2=20$
$\Rightarrow\text{AB}=2\sqrt{5}\text{ units}$
Now, $\text{AB}=\frac{2}{9}\text{AC}$
R. H. S $=\frac{2}{9}\times9\sqrt{5}$
$=2\sqrt{5}$
$=\text{AB}$
Hence, $\text{AB}=\frac{2}{9}\text{AC}$ is true.

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