Home
Class 12
MATHS
In an acute triangle A B C , if the coor...

In an acute triangle `A B C` , if the coordinates of orthocentre `H` are `(4,b)` , of centroid `G` are `(b ,2b-8)` and of circumcenter `S` are `(-4,8)` , then `b` cannot be (a) `4` (b) `8` (c) 12 (d) `-12`

A

(a) 4

B

(b) 8

C

(c) 12

D

(d) `-12`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
As H (orhtocenter), G (centroid), and C (circumcenter) are collinear we have
`|{:(4,,b,,1),(b,,2b-8,,1),(-4,,8,,1):}|=0`
or `|{:(4,,b,,1),(b-4,,b-8,,0),(-(b+4),,16-2b,,0):}|=0`
or `(b-4)(16-2b)+(b+4)(b-4)=0`
or `2(b-4)(8-b)+(b+4)+(b-8)=0`
or `(8-b)[2b-8)-(b+4)=0`
or `(8-b)(b-12)=0`
Hence `b=8 or 12`, which is wrong because collinearity does not explain centroid, orthocenter, and circumcenter.
Now, H.G, and C are collinear and `HG//GC=2`. Therefore,
`(-8+4)/(3)=b or b=(-4)/(3)`
and `(16+b)/(3)=2b-8 or b=8`
But no common value of b is possible.
Promotional Banner

Topper's Solved these Questions

  • COORDINATE SYSYEM

    CENGAGE PUBLICATION|Exercise Linked|10 Videos

  • COORDINATE SYSYEM

    CENGAGE PUBLICATION|Exercise Matrix match type|4 Videos

  • COORDINATE SYSYEM

    CENGAGE PUBLICATION|Exercise Exercises|59 Videos

  • COORDINATE SYSTEM

    CENGAGE PUBLICATION|Exercise Multiple Correct Answers Type|2 Videos

  • CROSS PRODUCTS

    CENGAGE PUBLICATION|Exercise DPP 2.2|13 Videos

Similar Questions

Explore conceptually related problems

In A B C , if the orthocentre is (1,2) and the circumcenter is (0, 0), then centroid of A B C) is (a) (1/2,2/3) (b) (1/3,2/3) (c) (2/3,1) (d) none of these

In a triangle ABC, coordinates of A are (1,2) and the equations of the medians through B and C are x+y=5 and x=4 respectively. Find the coordinates of B and C.

Two sides of a triangle are parallel to the coordinate axes. If the slopes of the medians through the acute angles of the triangle are 2 and m , then m is (a) 1/2 (b) 2 (c) 4 (d) 8

the coordinates of the vertices A,B,C of the triangle ABC are (7,-3) , (x,8) and (4,y) respectively , if the coordinates of the centroid of the triangle be (2,-1) , find x and y.

The co-ordinattes of B and C of the trianglle ABC are (5,2,8) and (2,-3,4) respectively.If the centroid of the triangle ABC are (3,-1,3) then the coordinates of A are ___

Using integration find the area of the triangle whose vertices are A(-4,3) ,B ( 3,4) and C( 8,6)

In triangle A B C , if sinAcosB=1/4 and 3t a n A=t a n B ,t h e ncot^2A is equal to (a)2 (b) 3 (c) 4 (d) 5.

In any triangle ABC, If b^(2) = a(c+a), c^(2) = b(a+b) , then prove that, cosAcosB cosC =-1/8 .

If a ,b and c are in A.P. and b-a ,c-b and a are in G.P., then a : b : c is (a). 1:2:3 (b). 1:3:5 (c). 2:3:4 (d). 1:2:4

In triangle A B C ,b^2sin2C+c^2sin2B=2bc where b=20 ,c=21 , then inradius= (a) 4 (b) 6 (c) 8 (d) 9