Home
Class 12
MATHS
if (x(1),x(2))^(2)+(y(1)-y(2))^(2)=a^(2)...

if `(x_(1),x_(2))^(2)+(y_(1)-y_(2))^(2)=a^(2), (x_(2)-x_(3))^(2)+(y_(2)-y_(3))^(2)=b^(2) (x_(3)-x_(1))^(2)+(y_(3)-y_(1))^(2)=c^(2).` where a,b,c are positive then prove that `4 |{:(x_(1),,y_(1),,1),(x_(2) ,,y_(2),,1),( x_(3),, y_(3),,1):}| = (a+b+c) (b+c-a)
(c+a-b)(a+b-c)`

Text Solution

Verified by Experts

we have `sqrt((x_(1)-x_(2))^(2)+(y_(1)-y_(2))^(2))=a`
`sqrt((x^(2)-x_(3))^(2)+(y_(2)-y_(3))^(2))=b`
`sqrt((x_(3)-x_(1))^(2)+(y_(3)-y_(1))^(2))=c`
Consider triangle having vertices `A(x_(1),y_(1)),B(x_(2),y_(2))` and
So , `AB=a,BC=b " and "AC= c.Also ,2s=a+b+c` where s is semi-perimetre.
Area of triangle ABC
`Delta=(1)/(2) ||{:(x_(1),,y_(1),,1),(x_(2),,y_(2),,1),(x_(3),,y_(3),,1):}||`
Also `Delta= sqrt(s(s-a)(s-b)(s-c))`
`:. ||{:(x_(1),,y_(1),,1),(x_(2),,y_(2),,1),(x_(3),,y_(3),,1):}||=sqrt(s(s-a)(s-b)(s-c))`
Squaring and simplifying we get
`4 |{:(x_(1),,y_(1),,1),(x_(2),,y_(2),,1),(x_(3),,y_(3),,1):}|^(2)`
=`(a+b+c)(b+c-a)(c+a-b)(a+b-c)`
Promotional Banner

Topper's Solved these Questions

  • DETERMINANT

    CENGAGE|Exercise Multiple Correct Answer|5 Videos

  • DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS

    CENGAGE|Exercise Exercise|337 Videos

Similar Questions

Explore conceptually related problems

it x_(1)^(2) +2y_(1)^(2)+3z_(1)^(2)=x_(2)^(2)+2y_(2)^(2)+3z_(2)^(2)=x_(3)^(2)+2y_(3)^(2)+3z_(3)^(2)=2 " and " x_(2)x_(3) +2y_(2)y_(3)+3z_(2)z_(3)=x_(3)x_(1)+2y_(3)y_(1)+3z_(3)z_(1)=x_(1)x_(2)+2y_(1)y_(2)+3z_(1)z_(2)=1 Then find the value of |{:(x_(1),,y_(1),,z_(1)),(x_(2),,y_(2),,z_(2)),(x_(3),,y_(3),,z_(3)):}|

If =|(2a,x_(1),y_(1)),(2b,x_(2),y_(2)),(2c,x_(3),y_(3))|=abc/2 ne 0 then the area area of the triangle whose vertices are (x_(1)/a,y_(1)/a),(x_(2)/b,y_(2)/b),(x_(3)/c,y_(3)/c) is:

If x_(1), x_(2), x_(3) and y_(1),y_(2), y_(3) are in arithmetic progression with the same common difference then the points (x_(1),y_(1)) (x_(2),y_(2)) (x_(3), y_(3)) are:

If x_(1),x_(2),x(3) as well as y_(1),y_(2),y_(3) are in geometric progression with the same common ratio,then the points (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3))

If (x_1-x_2)^2+(y_1-y_2)^2=a^2 , (x_2-x_3)^2+(y_2-y_3)^2=b^2 , (x_3-x_1)^2+(y_3-y_1)^2=c^2 , and k|[x_1,y_1, 1],[x_2,y_2, 1],[x_3,y_3, 1]|=(a+b+c)(b+c-b)(c+a-b)xx(a+b-c) , then the value of k is 1 b. 2 c. 4 d. none of these

If three point A(x_(1), y_(1)), B(x_(2), y_(2)) and C(x_(3), y_(3)) will be collinear then the area of triangleABC= ____.

Given (x_(1) + iy_(1)) (x_(2) + iy_(2))…(x_(n) + iy_(n)) = a + ib, show that (x_(1)^(2)+y_(1)^(2))(x_(2)^(2) + y_(2)^(2))(x_(3)^(2) + y_(3)^(2))…(x_(n)^(2) + y_(n)^(2)) = a^(2) + b^(2)

Prove that |(x+2a,y+2b,z+2c),(x,y,z),(a,b,c)|=0

if x,y and z are not all zero and connected by the equations a_(1)x+b_(1)y+c_(1)z=0,a_(z)x+b_(2)y+c_(2)z=0 and (p_(1)+lambdaq_(1))x+(p_(2)+lambdaq_(2))y+(p_(3)+lambdaq_(3))z=0 show that lambda =-|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(p_(1) ,, p_(2),,p_(3)):}|-:|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(q_(1) ,, q_(2),,q_(3)):}|

If S_(n)=(x+y)+(x^(2)+xy+y^(2))+(x^(3)+x^(2)y+y^(2)x+y^(3))+…n terms then prove that (x-y)S_(n)=[(x^(2)(x^(n)-1))/(x-1)-(y^(2)y^(n)-1)/(y-1)] .

CENGAGE-DETERMINANTS-All Questions
  1. if (x(1),x(2))^(2)+(y(1)-y(2))^(2)=a^(2), (x(2)-x(3))^(2)+(y(2)-y(3))^...

    Text Solution

    |

  2. Statement 1: If b c+q r=c a+r p=a b+p q=-1, t h e n|a p a p b q b q c...

    Text Solution

    |

  3. If f(theta)=|sinthetacosthetasintheta costhetasinthetacostheta costhet...

    Text Solution

    |

  4. If f(theta)=|[sin^2A,cot A,1],[sin^2B,cosB,1],[sin^2C,cosC,1]| , then...

    Text Solution

    |

  5. The roots of the equation |^x Cr^(n-1)Cr^(n-1)C(r-1)^(x+1)Cr^n Cr^n C(...

    Text Solution

    |

  6. Let Delta(x)=|[3,3x,3x^2+2a^2] , [3x, 3x^2+2a^2, 3x^3+6a^2x] , [3x^2+2...

    Text Solution

    |

  7. The determinant "Delta"=|(a^2+x) ab ac ab (b^2+x) bc ac bc (c^2+...

    Text Solution

    |

  8. If f(x)=|[a,-1, 0],[a x, a,-1],[a x^2,a x, a]|,t h e nf(2x)-f(x) is d...

    Text Solution

    |

  9. If g(x)-(f(x))/((x-a)(x-b)(x-c)),w h e r ef(x) is a polynomial of d...

    Text Solution

    |

  10. If (x)= |x^2+4x-3 2x+4 13 ...

    Text Solution

    |

  11. If |y z-x^2z x-y^2x y-z^2x z-y^2x y-z^2y z-x^2x y-z^2y z-x^2z x-y^2|=|...

    Text Solution

    |

  12. Let f (n) = |{:(n,,n+1,,n+2),(.^(n)P(n),,.^(n+1)P(n+1),,.^(n+2)P(n+2)...

    Text Solution

    |

  13. If |x^n x^(n+2)x^(2n)1x^a a x^(n+5)x^(a+6)x^(2n+5)|=0,AAx in R ,w h e...

    Text Solution

    |

  14. Let x<1, then value of [[x^2+2, 2x+1 ,1],[ 2x+1,x+2, 1],[ 3, 3 ,1]...

    Text Solution

    |

  15. If f (x) = |(0,x-a,x-b),(x+a,0,x-c),(x+b,x+c,0)|then

    Text Solution

    |

  16. Value of [[x+y, z,z ],[x, y+z, x],[y, y, z+x]], where x ,y ,z are non...

    Text Solution

    |

  17. Which of the following is not the root of the equation |[x,-6,-1],[ 2,...

    Text Solution

    |

  18. If A,B,C are the angles of a non right angled triangle ABC. Then find ...

    Text Solution

    |

  19. If f(x)=|x a a a x a a a x|=0, then f^(prime)(x)=0a n df^(x)=0 has com...

    Text Solution

    |

  20. Evaluate {:[( cos alpha cos beta , cos alpha sin beta , -sin alpha ),...

    Text Solution

    |

  21. Roots of the equation |x m n1a x n1a b x1a b c1|=0 are independent of ...

    Text Solution

    |