JEE 2023 — Mathematics PYQ

Step 1: Orthocenter $(\alpha ,\beta )$ ke liye equations banana

Orthocenter wo point hai jahan altitudes milte hain.

1. Altitude from $A$ to $BC$:

   • Slope of $BC=\frac{5-2}{4-(-1)}=\frac{3}{5}$

   • Slope of altitude $AD=-\frac{5}{3}$

   • Equation: $y-(-7)=-\frac{5}{3}(x-3)⟹3y+21=-5x+15⟹5x+3y=-6$ --- (i)

2. Altitude from $B$ to $AC$:

   • Slope of $AC=\frac{5-(-7)}{4-3}=\frac{12}{1}=12$

   • Slope of altitude $BE=-\frac{1}{12}$

   • Equation: $y-2=-\frac{1}{12}(x-(-1))⟹12y-24=-x-1⟹x+12y=23$ --- (ii)

Step 2: $\alpha$ aur $\beta$ ki value nikalna

Equations (i) aur (ii) ko solve karne par:

• Equation (ii) se $x=23-12y$. Isse (i) mein rakhne par:

• $5(23-12y)+3y=-6⟹115-60y+3y=-6⟹-57y=-121⟹\beta =\frac{121}{57}$

• $x=23-12(\frac{121}{57})=\frac{1311-1452}{57}=-\frac{141}{57}⟹\alpha =-\frac{47}{19}$

Step 3: Final Expression

Hame $9\alpha -6\beta +60$ nikalna hai:

Sahi vikalp (4) hai.