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`

Three points P(h,k), Q(x_(1) , y_(1) ) and R(x_(2) , y_(2) ) lie on a line. Show that, (h-x_(1) ) (y_(2) -y_(1) ) = (k-y_(1) ) ( x_(2) - x_(1) ) .

An equilateral triangle has each side equal 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 determinant |(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1)| equals

If the co-ordinates of the vertices of an equilateral trianlg with sides of length 'a' are (x_1,y_1),(x_2,y_2),(x_3,y_3) , then Prove that |{:(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1):}|^2=(3/4)a^4.

If sum_(i=1)^4(x_i^2+y_ i^2)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 (a) the vertices of a rectangle (b)collinear (c)the vertices of a trapezium (d)none of these

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 its vertices, then the value of the determinant |x_1y_1 1x_2y_2 1x_3y_3 1|^2 is equal to: 192 (b) 243 (c) 486 (d) 972

If P(x, y) is a point equidistant from the points A(6, -1) and B(2, 3), show that x-y = 3.

If x=sint,y=sinKt then show that (1-x^(2))y_2-xy_(1)+K^(2)y=0 .

If y= 3 cos (log x) + 4 sin (log x) , show that x^(2)y_(2) + xy_(1) + y= 0

Solve by the elimination method: (x+1)/(2)+(y-1)/(3)=9 and (x+1)/(3)+(y-1)/(2)=8