A line of slope `lambda(0 lt lambda lt 1)` touches the parabola `y+3x^2=0` at `P`. If `S` is the focus and `M` is the foot of the perpendicular of directrix from `P` , then `tan/_M P S`

Find the coordinates of the foot of the perpendicular P from the origin to the plane 2x-3y+4z-6=0

The circle x^2+y^2+2lambdax=0,lambda in R , touches the parabola y^2=4x externally. Then, (a) lambda>0 (b) lambda 1 (d) none of these

If the vertex of the parabola is (3,2) and directrix is 3x+4y-(19)/7=0 , then find the focus of the parabola.

The distance of two points P and Q on the parabola y^(2) = 4ax from the focus S are 3 and 12 respectively. The distance of the point of intersection of the tangents at P and Q from the focus S is

If P be a point on the parabola y^2=3(2x-3) and M is the foot of perpendicular drawn from the point P on the directrix of the parabola, then length of each sides of an equilateral triangle SMP(where S is the focus of the parabola), is

The foot of the perpendicular from A (1, 0, 0) to the line (x-1)/2=(y+1)/(-3)=(z+10)/8 is

If line x-2y-1=0 intersects parabola y^(2)=4x at P and Q, then find the point of intersection of normals at P and Q.

Two tangents on a parabola are x-y=0 and x+y=0. S(2,3) is the focus of the parabola. The length of latus rectum of the parabola is

let L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance from the two planes P_1 : x + 2y-z +1 = 0 and P_2 : 2x-y + z-1 = 0 , Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane P_1 . Which of the following points lie(s) on M?

y = sqrt(3)x +lambda is drawn through focus S of the parabola y^(2)= 8x +16 . If two intersection points of the given line and the parabola are A and B such that perpendicular bisector of AB intersects the x-axis at P then length of PS is