JEE Mathematics 2023 Question with Solution

Given: The triangle has vertices $A(3,-7)$, $B(-1,2)$ and $C(4,5)$.

Find: The value of $9\alpha-6\beta+60$, where $(\alpha,\beta)$ is the orthocenter.

Find the slopes of two sides and then use perpendicular slopes to write the equations of two altitudes.

Slope of $BC$ is

$$m_{BC}=\frac{5-2}{4-(-1)}=\frac{3}{5}$$

So, the slope of the altitude from $A$ is

$$m_1=-\frac{5}{3}$$

Slope of $AC$ is

$$m_{AC}=\frac{5-(-7)}{4-3}=12$$

So, the slope of the altitude from $B$ is

$$m_2=-\frac{1}{12}$$

Equation of altitude from $A(3,-7)$:

$$y+7=-\frac{5}{3}(x-3)$$ $$3y+21=-5x+15$$ $$5x+3y+6=0$$

Equation of altitude from $B(-1,2)$:

$$y-2=-\frac{1}{12}(x+1)$$ $$12y-24=-x-1$$ $$x+12y-23=0$$

The orthocenter is the intersection of these two altitudes. So solve

$$5x+3y=-6$$ $$x+12y=23$$

Multiply the second equation by $5$:

$$5x+60y=115$$

Subtract the first equation:

$$57y=121$$ $$y=\frac{121}{57}$$

Substitute back into $x+12y=23$:

$$x+12\cdot\frac{121}{57}=23$$ $$x=\frac{57\cdot 23-1452}{57}=-\frac{47}{57}$$

Thus,

$$\alpha=-\frac{47}{57},\quad \beta=\frac{121}{57}$$

Now compute the required expression:

$$9\alpha-6\beta+60=9\left(-\frac{47}{57}\right)-6\left(\frac{121}{57}\right)+60$$ $$=\frac{-423-726+3420}{57}=\frac{2271}{57}=25$$

Therefore, the value of the expression is $25$, so the correct option is A.

Given: The orthocenter is the intersection point of altitudes.

Find: The required linear expression in $\alpha$ and $\beta$.

A shortcut here is that only two altitudes are needed to locate the orthocenter. There is no need to use all three sides.

• Use side $BC$ to get the altitude from $A$.
• Use side $AC$ to get the altitude from $B$.
• Solve the two linear equations directly.

This works because all altitudes of a triangle are concurrent at the orthocenter.

From the working:

$$5x+3y+6=0$$ $$x+12y-23=0$$

Their intersection gives

$$(\alpha,\beta)=\left(-\frac{47}{57},\frac{121}{57}\right)$$

Then substitute into

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

to get $25$. Hence the correct option is A.