Straight Lines Question 9
Question 9 - 01 February - Shift 1
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
(1) $x^{2}-19 x+90=0$
(2) $x^{2}-18 x+80=0$
(3) $x^{2}-22 x+120=0$
(4) $x^{2}-20 x+99=0$
Show Answer
Answer: (4)
Solution:
Formula: Orthocentre Formula, Condition for Perpendicular Lines
Here $mBH \times mAC=-1$
$(\frac{\beta-3}{\alpha-2})(\frac{1}{-2})=-1$
$\beta-3=2 \alpha-4$
$\beta=2 \alpha-1$
$m _{AH} \times m _{BC}=-1$
$\Rightarrow \quad(\frac{\beta-2}{\alpha-1})(-2)=-1$
$\Rightarrow \quad 2 \beta-4=\alpha-1$
$\Rightarrow 2(2 \alpha-1)=\alpha+3$
$\Rightarrow 3 \alpha=5$
$\alpha=\frac{5}{3}, \beta=\frac{7}{3} \Rightarrow H(\frac{5}{3}, \frac{7}{3})$
$\alpha+4 \beta=\frac{5}{3}+\frac{28}{3}=\frac{33}{3}=11$
$\beta+4 \alpha=\frac{7}{3}+\frac{20}{3}=\frac{27}{3}=9$
$x^{2}-20 x+99=0$
