Question

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

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

Answer 1

Answer: (B)

$ \left(\frac{\beta-3}{\alpha-2}\right)\left(\frac{1}{-2}\right)=-1 $
$ \beta-3=2 \alpha-4 $
$ \beta=2 \alpha-1 $
$ m_{ AH } \times m_{ BC }=-1$
$\Rightarrow \quad\left(\frac{\beta-2}{\alpha-1}\right)(-2)=-1$
$ \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} $
$\Rightarrow H \left(\frac{5}{3}, \frac{7}{3}\right)$
$ \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 $
$ x^2-20 x+99=0$