Home
Class 12
MATHS
If A(x1,y1),B(x2,y2), and C(x3,y3) are t...

If `A(x_1,y_1),B(x_2,y_2),` and `C(x_3,y_3)` are three non-collinear points such that `x_1^2+y_1^2=x_2^2+y_2^2=x_3^2+y_3^2,` then prove that `x_1sin2A+x_2sin2B+x_3sin2C=y_1sin2A+y_2sin2B+y_3sin2C=0.`

Text Solution

Verified by Experts

Since `x_(1)^(2)+y_(1)^2=x_(2_^2=x_(3)^2`, circumcentre of `DeltaABC` is `(0,0)`.
Now, coordinates of circumcentre are
`((x_1sin2A+x_2sin2B+x_3sin2C)/(sin2A+sin2B+sin2C),(y_1sin2A+y_2sin2b+y_3sin2C)/(sin2A+sin2B+sin2C))=(0,0)`
Therefore,`x_1sin2A+x-2sin2B+x_3sin2C=y_1sin2A+y_2sin2B+y_3sin2C=0`
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • COORDINATE SYSYEM

    CENGAGE PUBLICATION|Exercise Illustration1.31|1 Videos

  • COORDINATE SYSYEM

    CENGAGE PUBLICATION|Exercise Illustration1.32|1 Videos

  • COORDINATE SYSYEM

    CENGAGE PUBLICATION|Exercise Illustration1.29|1 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

If A(x_1,y_1),B(x_2,y_2) and C(x_3,y_3) are the vertices of traingle ABC and x_(1)^(2)+y_(1)^(2)=x_(2)^(2)+y_(2)^(2)=x_3^(2)+y_(3)^(2) , then show that x_1 sin2A+x_2sin2B+x_3sin2C=y_1sin2A+y_2sin2B+y_3sin 2C=0 .

If the points (x_1, y_1),(x_2,y_2), and (x_3, y_3) are collinear show that (y_2-y_3)/(x_2x_3)+(y_3-y_1)/(x_3x_1)+(y_1-y_2)/(x_1x_2)=0

Knowledge Check

  • If x_1, x_2 , x_3 and y_1 , y_2 , y_3 are both in G.P. with the same common ratio, then the points (x_1 , y_1),(x_2 , y_2) and (x_3 , y_3)

    A
    are vertices of a triangle
    B
    lie on a straight line
    C
    lie on an ellipse
    D
    lie on a circle

    • Similar Questions

    Explore conceptually related problems

    If (x_i ,y_i),i=1,2,3, are the vertices of an equilateral triangle such that (x_1+2)^2+(y_1-3)^2=(x_2+2)^2+(y_2-3)^2=(x_3+2)^2+(y_3-3)^2, then find the value of (x_1+x_2+x_3)/(y_1+y_2+y_3) .

    If A (x_1,y_1) , B(x_2,y_2) and C (x_3,y_3) are vertices of an equilateral triangle whose each side is equal to a unit , then prove that, |{:(x_1,y_1,2),(x_2,y_2,2), (x_3,y_3,2):}|^2=3a^4

    If y=sin^-1x , show that, (1-x^2)y_2-xy_1=0 .

    If the join of (x_1,y_1) and (x_2,y_2) makes on obtuse angle at (x_3,y_3), then prove that (x_3-x_1)(x_3-x_2)+(y_3-y_1)(y_3-y_2)<0

    If y^(1/3) + y^(-1/3) = 2x, prove that (x^2 - 1)y_2 + xy_1 = 9y .

    If the normals to the ellipse x^2/a^2+y^2/b^2= 1 at the points (x_1, y_1), (x_2, y_2) and (x_3, y_3) are concurrent, prove that |(x_1,y_1,x_1y_1),(x_2,y_2,x_2y_2),(x_3,y_3,x_3y_3)|=0 .

    If x^2=y^3 , then prove that (x/y)^(3/2)+(y/x)^-(2/3)=x^(1/2)+y^(1/3) .

    Home
    Profile