Home
Class 12
MATHS
If the coordinates of the vertices of an...

If the coordinates of the vertices of an equilateral triangle with sides of length `a` are `(x_(1),y_(1)),(x_(2),y_(2))` and `(x_(3),y_(3))` then `|(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1)|^2=(3a^(4))/4`

Text Solution

Verified by Experts

The correct Answer is:
N/a
Since we know that area of a triangle with vertices `(x_(1),y_(1)),(x_(2),y_(2))` and `(x_(3),y_(3))` is given by
`Delta=1/2|(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1)|`
`impliesDelta^(2)=1/4|(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1)|^(2)`...................(i)
We know that ara fo an equilateral triangle with side `a`.
`=Delta=1/2((sqrt(3))/2)a^(2)=(sqrt(3))/4a^(2)`
`implies Delta^(2)=3/16a^(4)`.............(ii)
From eqs (i) and (ii) `3/16 a^(4)=1/4 |(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1)|^(2)`
`implies |(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1)|^(2)=3/4a^(4)` Hence proved
Promotional Banner

Topper's Solved these Questions

  • DETERMINANTS

    NCERT EXEMPLAR ENGLISH|Exercise LONG ANSWER TYPE QUESTIONS|6 Videos

  • DETERMINANTS

    NCERT EXEMPLAR ENGLISH|Exercise OBJECTIVE TYPE QUESTIONS|14 Videos

  • CONTINUITY AND DIFFERENTIABILITY

    NCERT EXEMPLAR ENGLISH|Exercise True/False|10 Videos

  • DIFFERENTIAL EQUATIONS

    NCERT EXEMPLAR ENGLISH|Exercise Objective|1 Videos

Similar Questions

Explore conceptually related problems

An equilateral triangle has each of its sides of length 4 cm. If (x_(r),y_(r)) (r=1,2,3) are its vertices the value of |{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|^2

An equilateral triangle has each side to a. If the coordinates of its vertices are (x_(1), y_(1)), (x_(2), y_(2)) and (x_(3), y_(3)) then the square of the determinat |(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1)| equals

An equilateral triangle has each of its sides of length 6 cm. If (x_(1),y_(1)), (x_(2),y_(2)) & (x_(3),y_(3)) are the verticles, then the value of the determinant |{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|^(2) is equal to :

Find the coordinates of the centroid of the triangle whose vertices are (x_1,y_1,z_1) , (x_2,y_2,z_2) and (x_3,y_3,z_3) .

The points on the curve f(x)=(x)/(1-x^(2)) , where the tangent to it has slope equal to unity, are (x_(1), y_(1)), (x_(2), y_(2)) and (x_(3), y_(3)) . Then, x_(1)+x_(2)+x_(3) is equal to

If (x_(1),y_(1)) and (x_(2),y_(2)) are ends of focal chord of y^(2)=4ax then x_(1)x_(2)+y_(1)y_(2) =

STATEMENT-1: If three points (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)) are collinear, then |{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|=0 STATEMENT-2: If |{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|=0 then the points (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)) will be collinear. STATEMENT-3: If lines a_(1)x+b_(1)y+c_(1)=0,a_(2)=0and a_(3)x+b_(3)y+c_(3)=0 are concurrent then |{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=0

If A(x_(1), y_(1)), B(x_(2), y_(2)) and C (x_(3), y_(3)) are the vertices of a Delta ABC and (x, y) be a point on the internal bisector of angle A, then prove that b|(x,y,1),(x_(1),y_(1),1),(x_(2),y_(2),1)|+c|(x,y,1),(x_(1),y_(1),1),(x_(3),y_(3),1)|=0 where, AC = b and AB = c.

Write the formula for the area of the triangle having its vertices at (x_1,\ y_1),\ \ (x_2,\ y_2) and (x_3,\ y_3) .

Show that the coordinates off the centroid of the triangle with vertices A(x_1, y_1, z_1),\ B(x_2, y_2, z_2)a n d\ (x_3, y_3, z_3) are ((x_1+x_2+x_3)/3,(y_1+y_2+y_3)/3,(z_1+z_2+z_3)/3)