$(-2,-1),(2,5)$ are two vertices of a triangle and $\left(2, \frac{5}{3}\right)$ is its orthocenter. If $(m, n)$ is the third vertex of that triangle then $m+n=$

• A

  -4
• B

  -2
• Correct answer
  C

  5
• D

  8

Solution

Shope of $A F=\frac{\frac{5}{3}+1}{2+2}=\frac{8}{12}=\frac{2}{3}$ $\therefore$ Slope of $B C=\frac{-3}{2}$ Equation of $B C: y-5=\frac{-3}{2}(x-2) \Rightarrow 3 x+2 y-16=0 \quad ...\mathrm{(i)}$ Slope of $B G=\frac{\frac{5}{3}-5}{2-2}=\frac{1}{0}$ $\therefore$ Siope of $A C=0$ Equation of $A C: y+1=0(x+2) \Rightarrow y=-1$ From (i) $x=6$ $\therefore m+n=6-1=5 .$

Hence, the correct answer is option: 'C'.