JEE MAIN - Mathematics (2022 - 27th July Evening Shift - No. 10)
The equations of the sides $$\mathrm{AB}, \mathrm{BC}$$ and CA of a triangle ABC are $$2 x+y=0, x+\mathrm{p} y=39$$ and $$x-y=3$$ respectively and $$\mathrm{P}(2,3)$$ is its circumcentre. Then which of the following is NOT true?
$$(\mathrm{AC})^{2}=9 \mathrm{p}$$
$$(\mathrm{AC})^{2}+\mathrm{p}^{2}=136$$
$$32<\operatorname{area}\,(\Delta \mathrm{ABC})<36$$
$$34<\operatorname{area}\,(\triangle \mathrm{ABC})<38$$
Explanation
Intersection of $$2x + y = 0$$ and $$x - y = 3\,:\,A(1, - 2)$$
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Equation of perpendicular bisector of AB is
$$x - 2y = - 4$$
Equation of perpendicular bisector of AC is
$$x + y = 5$$
Point B is the image of A in line $$x - 2y + 4 = 0$$ which is obtained as $$B\left( {{{ - 13} \over 5},{{26} \over 5}} \right)$$
Similarly vertex $$C:(7,4)$$
Equation of line $$BC:x + 8y = 39$$
So, $$p = 8$$
$$AC = \sqrt {{{(7 - 1)}^2} + {{(4 + 2)}^2}} = 6\sqrt 2 $$
Area of triangle $$ABC = 32.4$$
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