Animated Solution for Mathematics - Straight Lines: 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

Imagine you are standing on a coordinate plane, looking at a triangle defined by vertices $A(3,-7)$, $B(-1,2)$, and $C(4,5)$. Our mission is to find the orthocenter, the elusive point where the three altitudes of this triangle meet.

Because angle $B$ is obtuse, our orthocenter will not be nestled inside the triangle; it will be out in the open, beyond the edges. This is the first lesson of coordinate geometry: trust the math, even when your intuition says the point should be inside.

Perpendicularity

To find the orthocenter $H(\alpha ,\beta )$, we do not need to hunt down all three altitudes. The intersection of any two is sufficient to define a unique point. We will focus on the altitude from $A$ (perpendicular to $BC$) and the altitude from $B$ (perpendicular to $AC$).

First, we find the slope of side $BC$, denoted as $m_{BC}$. Using the formula $\frac{y_2-y_1}{x_2-x_1}$, we calculate:

Since the altitude from $A$ is perpendicular to $BC$, its slope $m_1$ must be the negative reciprocal: $m_1=-\frac{5}{3}$. Using the point-slope form with vertex $A(3,-7)$, we write:

Rearranging this, we arrive at our first line equation:

Now, we repeat the process for the altitude from $B$. We find the slope of side $AC$:

The altitude from $B$ is perpendicular to $AC$, so its slope $m_2$ is $-\frac{1}{12}$. Using point $B(-1,2)$, the equation becomes:

Multiplying by $12$ and rearranging, we get our second line equation:

We now have a system of two linear equations:

Solving for $x$ in the second equation gives $x=23-12y$. Substituting this into the first equation:

This simplifies to $115-60y+3y=-6$, which leads to $-57y=-121$, or:

Substituting $\beta$ back into $x=23-12y$:

The problem asks us to evaluate $9\alpha -6\beta +60$. By keeping our values in their raw fractional form, we avoid early rounding errors and set the stage for a beautiful simplification.

Substituting $\alpha =-\frac{47}{19}$ and $\beta =\frac{121}{57}$ (noting that $\frac{121}{57}=\frac{121}{3\times 19}$):

The entire expression collapses into the elegant final answer of 25. Remember, in JEE Advanced, the complexity of the numbers is often a distraction; stay calm, keep your fractions raw, and watch the math reveal its own simplicity.