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QJEEM23Ma9222: JEE Main 2023 Mathematics - Class 11 JEEM23Ma9222 Solution: If the orthocentre of the triangle, whose vertices are (1, 2),…

If the orthocentre of the triangle, whose vertices are (1, 2), (2, 3) and (3, 1) is (α,β)(\alpha,\beta) , then the quadratic equation whose roots are α+4β\alpha+4\beta and 4α+β4\alpha+\beta , is :

Multiple Choice Options

  • A x2−20x+99=0x^2-20x+99=0
  • B x2−22x+120=0x^2-22x+120=0
  • C x2−19x+90=0x^2-19x+90=0
  • D x2−18x+80=0x^2-18x+80=0

Step-by-Step Verified Solution

JEE Main 2023 (Online) 1st February Morning Shift Mathematics - Straight Lines and Pair of Straight Lines Question 44 English Explanation

m=−12\mathrm{m}=-\frac{1}{2}

Here mBH×mAC=−1\mathrm{m_BH} \times \mathrm{m_AC}=-1

amp;(β−3α−2)(1−2)=−1amp;β−3=2α−4amp;β=2α−1amp; mAH×mBC=−1⇒amp;(β−2α−1)(−2)=−1⇒amp;2β−4=α−1⇒amp;2(2α−1)=α+3⇒amp;3α=5amp;α=53,β=73⇒H(53,73)amp;α+4β=53+283=333=11amp;β+4α=73+203=273=9amp;x2−20x+99=0\begin{array}{ll} & \left(\frac{\beta-3}{\alpha-2}\right)\left(\frac{1}{-2}\right)=-1 \\\\& \beta-3=2 \alpha-4 \\\\& \beta=2 \alpha-1 \\\\& \mathrm{~m}_{\mathrm{AH}} \times \mathrm{m}_{\mathrm{BC}}=-1 \\\\\Rightarrow & \left(\frac{\beta-2}{\alpha-1}\right)(-2)=-1 \\\\\Rightarrow & 2 \beta-4=\alpha-1 \\\\\Rightarrow & 2(2 \alpha-1)=\alpha+3 \\\\\Rightarrow & 3 \alpha=5 \\\\& \alpha=\frac{5}{3}, \beta=\frac{7}{3} \Rightarrow \mathrm{H}\left(\frac{5}{3}, \frac{7}{3}\right) \\\\& \alpha+4 \beta=\frac{5}{3}+\frac{28}{3}=\frac{33}{3}=11 \\\\& \beta+4 \alpha=\frac{7}{3}+\frac{20}{3}=\frac{27}{3}=9 \\\\& \mathrm{x}^2-20 \mathrm{x}+99=0\end{array}

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