JEE MAIN - Mathematics (2022 - 29th July Evening Shift - No. 10)
Let $$\mathrm{A}(\alpha,-2), \mathrm{B}(\alpha, 6)$$ and $$\mathrm{C}\left(\frac{\alpha}{4},-2\right)$$ be vertices of a $$\triangle \mathrm{ABC}$$. If $$\left(5, \frac{\alpha}{4}\right)$$ is the circumcentre of $$\triangle \mathrm{ABC}$$, then which of the following is NOT correct about $$\triangle \mathrm{ABC}$$?
area is 24
perimeter is 25
circumradius is 5
inradius is 2
Explanation
_29th_July_Evening_Shift_en_10_1.png)
Circumcentre of $\triangle A B C$
$$ \begin{aligned} &=\left(\frac{\alpha+\frac{\alpha}{4}}{2}, \frac{6-2}{2}\right) \\\\ &=\left(\frac{5 \alpha}{8}, 2\right) \\\\ &=\left(5, \frac{\alpha}{4}\right) \\\\ &\Rightarrow \alpha=8 \end{aligned} $$
$\operatorname{area}(\triangle A B C)=\frac{1}{2} \cdot \frac{3 \alpha}{4} \times 8=24$ sq. units
$$ \begin{aligned} \text { Perimeter } &=8+\frac{3 \alpha}{4}+\sqrt{8^{2}+\left(\frac{3 \alpha}{4}\right)^{2}} \\\\ &=8+6+10=24 \end{aligned} $$
Circumradius $=\frac{10}{2}=5$
inradius $(r)=\frac{\Delta}{s}=\frac{24}{12}=2$
Comments (0)
