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 (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)

If x!=y!=za n d|[x,x^2, 1+x^3],[y ,y^2 ,1+y^3],[z, z^2, 1+z^3]|=0, then the value of x y z is a. 1 b. 2 c. -1 d. 2

If sin^2theta=(x^2+y^2+1)/(2x) , then x must be -3 (b) -2 (c) 1 (d) none of these

If a ,b , a n dc are in G.P. and x ,y , respectively, are the arithmetic means between a ,b ,a n db ,c , then the value of a/x+c/y is 1 b. 2 c. 1//2 d. none of these

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

If sum_(i-1)^4(xi2+y i2)lt=2x_1x_3+2x_2x_4+2y_2y_3+2y_1y_4, the points (x_1, y_1),(x_2,y_2),(x_3, y_3),(x_4,y_4) are the vertices of a rectangle collinear the vertices of a trapezium 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= ____.

Prove that |(a-x)^2(a-y)^2(a-z)^2(b-x)^2(b-y)^2(b-z)^2(c-x)^2(c-y)^2(c-z)^2|= |(1+a x)^2(1+b x)^2(1+c x)^2(1+a y)^2(1+b y)^2(1+c y)^2(1+a z)^2(1+b z)^2(1+c z)^2|= 2(b-c)(c-c)(a-b)xx(y-z)(z-x)(x-y)dot

If a(x^(2)+y^(2)+2y+1)=(x-2y+3)^(2) is an ellipse and a in (b, oo) , then the value of b is ___________