Find the equation of the straight lines passing through the following pair of point: `(a t_1, a//t_1)` and `(a t_2, a//t_2)`

Find the equation of the line passing through the point (-1,1) and (2,4).

Find the equations of the tangent and normal to the parabola y^2=4a x at the point (a t^2,2a t) .

Find the equation of the straight line, which passes through the point (1, 4) and is such that the segment of the line intercepted between the axes is divided by the point in the ratio 1: 2.

Find the vector equations of the plane passing through the points R(2,5,-3), S(-2,-3,5), T(5,3,-3)

Find the equations of the tangent and the normal to the following curves at the given point: x = a ( t + sin t ), y = a (1-cost) at t = pi/2

Find the equations of the tangent and the normal to the following curves at the given point: x = a cos t , y = b sin t at t = pi/2

Find the equations of the tangent and the normal to the following curves at the given point: x = sin 3t, y = cos 2 t at t = pi/4

Find the equation of the straight line which is a tangent at one point and normal at another point to the curve y=8t^(3)-1, x=4t^(2)+3 .

If t_1 a n d t_2 are roots of eth equation t^2+lambdat+1=0, where lambda is an arbitrary constant. Then prove that the line joining the points (a t1^2,2a t_1)a d n(a t2^2,2a t_2) always passes through a fixed point. Also, find the point.

Write the equation of a tangent to the curve x=t, y=t^2 and z=t^3 at its point M(1, 1, 1): (t=1) .