If the points $ \mathrm{A}(1,2), \mathrm{O}(0,0) $ and $ \mathrm{C}(a, b) $ are collinear, then
(A) $ a=b $
(B) $ a=2 b $
(C) $ 2 a=b $
(D) $ a=-b $
Given:
The points \( \mathrm{A}(1,2), \mathrm{O}(0,0) \) and \( \mathrm{C}(a, b) \) are collinear.
To do:
We have to choose the correct option.
Solution:
We know that,
If the points \( \mathrm{A}(1,2), \mathrm{O}(0,0) \) and \( \mathrm{C}(a, b) \) are collinear, then the area of triangle ABC is 0.
Area of a triangle $=\frac{1}{2}\left[x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right]$
Therefore,
Area of triangle ABC $=\frac{1}{2}[1(0-b)+0(b-2)+a(2-0)]$
$0=\frac{1}{2}[1(-b)+0+2a]$
$2(0)=2a-b$
$2a=b$
Hence, the correct option is (C) \( 2 a=b \).
Related Articles
- Identify the like termsa) \( -a b^{2},-4 b a^{2}, 8 a^{2}, 2 a b^{2}, 7 b,-11 a^{2},-200 a,-11 a b, 30 a^{2} \)\( \mathrm{b},-6 \mathrm{a}^{2}, \mathrm{b}, 2 \mathrm{ab}, 3 \mathrm{a} \)
- Draw any line segment, say \( \overline{\mathrm{AB}} \). Take any point \( \mathrm{C} \) lying in between \( \mathrm{A} \) and \( \mathrm{B} \). Measure the lengths of \( A B, B C \) and \( A C \). Is \( A B=A C+C B \) ?[Note : If \( A, B, C \) are any three points on a line such that \( A C+C B=A B \), then we can be sure that \( \mathrm{C} \) lies between \( \mathrm{A} \) and \( \mathrm{B} \).]
- In \( \Delta \mathrm{XYZ}, \quad \angle \mathrm{Y}=90^{\circ} \). If \( \mathrm{XY}=a^{2}-\mathrm{b}^{2} \) and \( \mathrm{YZ}=2 a b \), find \( \mathrm{XZ} .(a>b>0) \).
- If $a ≠ b ≠ c$, prove that the points $(a, a^2), (b, b^2), (c, c^2)$ can never be collinear.
- In \( \Delta \mathrm{PQR}, \mathrm{PD} \perp \mathrm{QR} \) such that \( \mathrm{D} \) lies on \( \mathrm{QR} \). If \( \mathrm{PQ}=a, \mathrm{PR}=b, \mathrm{QD}=c \) and \( \mathrm{DR}=d \), prove that \( (a+b)(a-b)=(c+d)(c-d) \).
- \( \mathrm{a}^{2}-\mathrm{b}^{2} \) is a product of ____ and ____.
- Find the values of \( k \) if the points \( \mathrm{A}(k+1,2 k), \mathrm{B}(3 k, 2 k+3) \) and \( \mathrm{C}(5 k-1,5 k) \) are collinear.
- If $a ≠ b ≠ 0$, prove that the points $(a, a^2), (b, b^2), (0, 0)$ are never collinear.
- If \( \triangle \mathrm{ABC} \) is right angled at \( \mathrm{C} \), then the value of \( \cos (\mathrm{A}+\mathrm{B}) \) is(A) 0(B) 1(C) \( \frac{1}{2} \)(D) \( \frac{\sqrt{3}}{2} \)
- If the points $A( -2,\ 1) ,\ B( a,\ b)$ and $C( 4,\ -1)$ are collinear and $a-b=1$, find the values of $a$ and $b$.
- In a parallelogram \( \mathrm{ABCD}, \angle \mathrm{A}: \angle \mathrm{B}=2: 3, \) then find angle \( \mathrm{D} \).
- If \( a+b=5 \) and \( a b=2 \), find the value of(a) \( (a+b)^{2} \)(b) \( a^{2}+b^{2} \)(c) \( (a-b)^{2} \)
- Factorize:$(a – b + c)^2 + (b – c + a)^2 + 2(a – b + c) (b – c + a)$
Kickstart Your Career
Get certified by completing the course
Get Started
Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies