Collinearity

Let $a_{1}, a_{2}, a_{3},....a_{n}$ and $b_{1} , b_{2}, b_{3},...b_{n}$ be two arithmetic progressions.

Then prove that the points : $(a_{1},b_{1} ),(a_{2},b_{2}),.......,(a_{n},b_{n})$ are collinear. Note by Aditya Parson
6 years, 5 months ago

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We will now show that any 3 points are collinear.

Thus, we have to show $(a_k, b_k)$, $(a_m, b_m)$ and $(a_n, b_n)$ are collinear.

$\iff$ Line formed by $(a_k, b_k)$ and $(a_m, b_m)$ is parallel to line formed by $(a_m, b_m)$ and $(a_n, b_n)$

Using the fact of arithmetic progressions, we can let $a_k=a_0+k \times d_a$ and $b_k=b_0+k \times d_b$, for some constants $a_0, d_a, b_0, d_b$

Therefore we can show that the gradients of the 2 lines are equal.

Gradient of line formed by $(a_k, b_k)$ and $(a_m, b_m)=\frac{b_m-b_k}{a_m-a_k}=\frac{d_b}{d_a}$

Gradient of line formed by $(a_m, b_m)$ and $(a_n, b_n)=\frac{b_m-b_n}{a_m-a_n}=\frac{d_b}{d_a}$

Since gradients are equal, it follows that the lines are parallel and thus $(a_k, b_k)$, $(a_m, b_m)$ and $(a_n, b_n)$ are collinear.

Now we can start with base case n=3 and induct by taking points $(a_1, b_1), (a_2, b_2)$ and $(a_{n+1}, b_{n+1})$. Hence proven.

- 6 years, 5 months ago

I will corroborate your solution there.

- 6 years, 5 months ago

Let $a_i = \lambda_a i+\mu_a$, $b_i = \lambda_b i+\mu_b$ for some real numbers $\lambda_a,\lambda_b,\mu_a,\mu_b$.

Then $i=\frac{a_i-\mu_a}{\lambda_a}$, and thus $b_i = \lambda_b\frac{a_i-\mu_a}{\lambda_a}+\mu_b = \frac{\lambda_b}{\lambda_a}a_i+(\mu_b-\frac{\mu_a\lambda_b}{\lambda_a})$.

Hence all the points $(a_i,b_i)$ lie on the same line $y=\frac{\lambda_b}{\lambda_a}x+(\mu_b-\frac{\mu_a\lambda_b}{\lambda_a})$, as desired.

- 6 years, 5 months ago

This prob came in C.M.I....................... Do you know how to prove that f(x)=1/(x+2cos(x)) is onto rigorously...I mean intuitively it can be seen, but is there any othere way to do it.............I mean using 'hardcore math'.

- 6 years, 5 months ago

There exists $f(x)$ whenever $x+2cosx \neq=0$. But I don't know how can we conclude that $x$ will never be $=-2cosx$ for all $x$.

- 6 years, 5 months ago

at x=-1........-x/2 and cos(x) are same.............so Domain,I guess, is R-{-1}......

- 6 years, 5 months ago

hey,you gave isi bmath exam,right...?

- 6 years, 5 months ago

Nope friend, B.Stat............what about you?

- 6 years, 5 months ago

there's a easy method using determinant criteria, for 3 points prove that the determinant of area equal to 0

- 6 years, 5 months ago

This method is far more convenient in exams, since it would take much less time. Determinant method is always applicable to show collinearity.

- 6 years, 5 months ago