If ( , ) is the orthocenter of the triangle ABC with vertices A(3,-7),B(-1,2) and C(4,5), …

Question

If $(\alpha ,\beta )$ is the orthocenter of the triangle $ABC$ with vertices $A(3,-7),B(-1,2)$ and $C(4,5)$, then $9\alpha -6\beta +60$ is equal to :

Accepted Answer

Altitude from $B$ perpendicular to $AC$ (slope of $AC=\dfrac{5+7}{4-3}=12$, so altitude slope $=-\dfrac1{12}$): $y-2=-\dfrac1{12}(x+1) \implies 12y-24=-x-1 \implies x+12y=23$

Altitude from $C$ perpendicular to $AB$ (slope of $AB=\dfrac{2+7}{-1-3}=-\dfrac94$, so altitude slope $=\dfrac49$): $y-5=\dfrac49(x-4) \implies 9y-45=4x-16 \implies$ ... following through the source's stated solving, $\alpha=-\dfrac{47}{19},\ \beta=\dfrac{121}{57}$.

$$9\alpha-6\beta+60 = 25$$

Solutions · 1