Home
Class 12
MATHS
In a A B C ,A-=(alpha,beta),B-=(1,2),C-...

In a ` A B C ,A-=(alpha,beta),B-=(1,2),C-=(2,3),` point `A` lies on the line `y=2x+3,` where `alpha,beta` are integers, and the area of the triangle is `S` such that `[S]=2` where `[` .`]` denotes the greatest integer function. Then the possible coordinates of `A` can be `(-7,-11)` (b) `(-6,-9)` `(2,7)` (d) `(3,9)`

A

`(-7,-11)`

B

`(-6,-9)`

C

(2,7)

D

(3,9)

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
The point `A(alpha,beta)` lies on `y=2x+3`.
Hence,
`beta=2alpha+3`
`A=(alpha, 2alpha+3)`
Area of `Delta ABC` is `:|(1)/(2)|:{:(alpha,,2alpha+3,,1),(1,,2,,1),(2,,3,,1):}:|":|`
`=|:(1)/(2)[alpha(2-3)+(2alpha+3)(2-1)+1(3-4)]:|`
`=(1)/(2)|-alpha+2alpha+3-1|=(1)/(2)|alpha+2|=S`
`[S]=2or 2leSlt3`
`therfore2le(1)/(2)|alpha+2|lt3`
`4le |alpha+2|lt6`
`|alpha+2|le6`
or `-6lealpha+2le6`
or `-8lealphale4`
and `|alpha+2|ge4`
i.e., `alpha+2ge4 or alpha+2le-4`
i.e., `alphage2 or alphale-6`
From (1) and (2) , we get
`-8ltalphalt-6or 2lealphalt4`
or `alpha=-7,-6,2,3`
The possible coordiantes of A are `(-7,-11) ,(-6,-9) ,(2,7)` and `(3,9)`.
Promotional Banner

Topper's Solved these Questions

  • COORDINATE SYSYEM

    CENGAGE|Exercise Exercise (Comprehension)|10 Videos

  • COORDINATE SYSYEM

    CENGAGE|Exercise Exercise (Matrix)|4 Videos

  • COORDINATE SYSYEM

    CENGAGE|Exercise Exercise (Single)|59 Videos

  • COORDINATE SYSTEM

    CENGAGE|Exercise Multiple Correct Answers Type|2 Videos

  • CROSS PRODUCTS

    CENGAGE|Exercise DPP 2.2|13 Videos

Similar Questions

Explore conceptually related problems

In a A B C ,A-=(alpha,beta),B-=(1,2),C-=(2,3), point A lies on the line y=2x+3, where alpha,beta are integers, and the area of the triangle is S such that [S]=2 where [ . ] denotes the greatest integer function. Then the possible coordinates of A can be (a) (-7,-11) (b) (-6,-9) (c) (2,7) (d) (3,9)

In a triangle A B C ,A=(alpha,beta)B=(2,3),a n dC=(1,3)dot Point A lies on line y=2x+3, where alpha in Idot The area of DeltaA B C , , is such that [Delta]=5 . The possible coordinates of A are (where [.] represents greatest integer function). (a) (2,3) (b) (5,13) (c) (-5,-7) (d) (-3,-5)

(lim)_(xvec(-1^)/3)1/x[(-1)/x]= (where [.] denotes the greatest integer function) a. -9 b. -12 c. -6 d. 0

The period of the function f(x)=(6x+7)+cospix-6x , where [dot] denotes the greatest integer function is: 3 (b) 2 pi (c) 2 (d) none of these

lim_(x-(pi)/(2)) [([sinx]-[cosx]+1)/(3)]= (where [.] denotes the greatest integer integer function)

Number of points of discontinuity of f(x)=[sin^(-1)x]-[x] in its domain is equal to (where [.] denotes the greatest integer function) a. 0 b. 1 c. 2 d. 3

T h ev a l u eofint_(-2)^1[x[1+cos((pix)/2)]+1]dx , where [.] denotes the greatest integer function, is 1 (b) 1//2 (c) 2 (d) none of these

The greatest value of f(x)=cos(x e^([x])+7x^2-3x),x in [-1,oo], is (where [.] represents the greatest integer function). -1 (b) 1 (c) 0 (d) none of these

The area of triangle A B C is 20c m^2dot The coordinates of vertex A are -5,0) and those of B are (3,0)dot The vertex C lies on the line x-y=2 . The coordinates of C are (5,3) (b) (-3,-5) (-5,-7) (d) (7,5)

The range of sin^(-1)[x^2+1/2]+cos^(-1)[x^2-1/2] , where [.] denotes the greatest integer function, is {pi/2,pi} (b) {pi} (c) {pi/2} (d) none of these