Để ba điểm M, N, và một điểm khác (ví dụ, P) thẳng hàng, vector $\vec{MN}$ và vector $\vec{MP}$ phải cùng phương, tức là tỉ lệ.
Ta có $\vec{MN} = (2-3; -5-(-1)) = (-1; -4)$.
Xét từng đáp án:
- A. P(0; 13): $\vec{MP} = (0-3; 13-(-1)) = (-3; 14)$. Kiểm tra tỉ lệ: $\frac{-3}{-1} = 3 \neq \frac{14}{-4} = -3.5$. Vậy P không thẳng hàng với M, N.
- B. Q(1; -8): $\vec{MQ} = (1-3; -8-(-1)) = (-2; -7)$. Kiểm tra tỉ lệ: $\frac{-2}{-1} = 2 \neq \frac{-7}{-4} = 1.75$. Vậy Q không thẳng hàng với M, N.
- B. Q(1; -8): corrected calculation $\vec{MQ} = (1-3, -8 - (-1)) = (-2, -7)$ This appears to be incorrect in the original calculation. However if we use $\vec{MQ}=(1-3;-8+1)=(-2;-7)$ then the ratio is $\frac{-2}{-1}=2$ and $\frac{-7}{-4}=\frac{7}{4}$ still not equal.
However if we instead consider $\vec{NQ}=(1-2;-8+5)=(-1;-3)$ then the ratio between this and $\vec{MN}=(-1;-4)$ is $\frac{-1}{-1}=1$ and $\frac{-3}{-4}=0.75$. Hence not equal. We were supposed to verify if the vectors are scalar multiples not compare the $\vec{MQ}$ from B with $\vec{MN}$.
$\frac{-1}{-1}=1 \frac{-3}{-4}=0.75$.
Consider calculating gradient between points: gradient between $M(3;-1)$ and $N(2;-5)$ is $\frac{-5+1}{2-3}=4$. The line equation is $y+1=4(x-3)$ so $y=4x-13$.
- A: $13 \neq 4(0)-13=-13$ not on the line
- B: $-8 \neq 4(1)-13=-9$ not on the line
- Incorrect point C and D are impossible to determine and are redundant.
- However $Q$ is closed $-9 \approx -8$ but is likely incorrect. $4x -13$. $-8 = 4x -13$, $4x=5$ so $x=1.25$ and its close!
Let's instead calculate equation from $M$ to $N$.
$M(3;-1)$ and $N(2;-5)$.
$ rac{y+1}{x-3} = rac{-5+1}{2-3}=4$. So $y=4x-13$.
Check B. $Q(1;-8)$, $-8 = 4(1)-13$ so $-8 = -9$. Not correct. However it is $Q(1;-9)$ that is in the line. So $Q$ is more close than A. I think this is wrong. lets check C and D
C: H(2;1) gradient with M(3;-1) is $\frac{1+1}{2-3} = -2$ not 4 so is wrong.
D: K(3;1), gradient with M is $\frac{1+1}{3-3}$ which is undefined. So is not equal.
There is an error. If we test Q(1;-9) Then $\vec{MQ}=(1-3;-9+1)=(-2;-8)$. $\vec{MN}=(-1,-4)$ then ratio is $2$ and $2$. so the answer is $Q(1;-9)$. But $-9$ is not any of the answers.
