Prove that the area of the triangle whose vertices are `(t ,t-2),(t+2,t+2),` and `(t+3,t)` is independent of `tdot`

Prove that the locus of the centroid of the triangle whose vertices are (acost ,asint),(bsint ,-bcost), and (1,0) , where t is a parameter, is circle.

Find the absolute value of parameter t for which the area of the triangle whose vertices the A(-1,1,2); B(1,2,3)a n dC(t,1,1) is minimum.

The slope of the tangent to the curve x= t^(2)+3t-8,y=2t^(2)-2t-5 at the point (2,-1) is

Prove that the segment of the normal to the curve x= 2a sin t + a sin t cos^2 t ; y = - a cos^3 t contained between the co-ordinate axes is equal to 2a.

Find the derivative of the function g(t) = ((t-2)/(2t+1)) .

The chord A B of the parabola y^2=4a x cuts the axis of the parabola at Cdot If A-=(a t_1 ^2,2a t_1),B-=(a t_2 ^2,2a t_2) , and A C : A B 1:3, then prove that t_2+2t_1=0 .

Find the equation of tangent and normal to the curve x=(2a t^2)/((1+t^2)),y=(2a t^3)/((1+t^2)) at the point for which t=1/2dot

Computing area with parametrically represented boundaries : If the boundary of a figure is represented by parametric equation, i.e., x=x(t), y=y(t), then the area of the figure is evaluated by one of the three formulas : S=-int_(alpha)^(beta)y(t)x'(t)dt, S=int_(alpha)^(beta)x(t)y'(t)dt, S=(1)/(2)int_(alpha)^(beta)(xy'-yx')dt, Where alpha and beta are the values of the parameter t corresponding respectively to the beginning and the end of the traversal of the curve corresponding to increasing t. The area of the loop described as x=(t)/(3)(6-t),y=(t^(2))/(8)(6-t) is

The vertices of a triangle are [a t_1t_2,a(t_1 +t_2)] , [a t_2t_3,a(t_2 +t_3)] , [a t_3t_1,a(t_3 +t_1)] Then the orthocenter of the triangle is (a) (-a, a(t_1+t_2+t_3)-at_1t_2t_3) (b) (-a, a(t_1+t_2+t_3)+at_1t_2t_3) (c) (a, a(t_1+t_2+t_3)+at_1t_2t_3) (d) (a, a(t_1+t_2+t_3)-at_1t_2t_3)