Let `S` be the focus of `y^2=4x` and a point `P` be moving on the curve such that its abscissa is increasing at the rate of 4 units/s. Then the rate of increase of the projection of `S P` on `x+y=1` when `P` is at (4, 4) is (a)`sqrt(2)` (b) `-1` (c) `-sqrt(2)` (d) `-3/(sqrt(2))`

A point P moves along the curve y=x^(3) . If its abscissa is increasing at the rate of 2 units/ sec, then the rate at which the slop of the tangent at P is increasing when P is at (1,1) , is

A point P moves along the curve y=x^(3) . If its abscissa is increasing at the rate of 2 units/ sec, then the rate at which the slop of the tangent at P is increasing when P is at (1,1) , is

A particle moves along the curve x^(2)=2y . At what point, ordinate increases at the same rate as abscissa increases ?

A normal drawn to the parabola y^2=4a x meets the curve again at Q such that the angle subtended by P Q at the vertex is 90^0dot Then the coordinates of P can be (a)(8a ,4sqrt(2)a) (b) (8a ,4a) (c)(2a ,-2sqrt(2)a) (d) (2a ,2sqrt(2)a)

A point P on the ellipse (x^(2))/(2)+(y^(2))/(1)=1 is at distance of sqrt(2) from its focus s. the ratio of its distance from the directrics of the ellipse is

The line x-y+2=0 touches the parabola y^2 = 8x at the point (A) (2, -4) (B) (1, 2sqrt(2)) (C) (4, -4 sqrt(2) (D) (2, 4)

An equilateral triangle SAB is inscribed in the parabola y^(2)=4ax having its focus at S. If chord AB lies towards the left of S, then the side length of this triangle is 2a(2-sqrt(3)) (b) 4a(2-sqrt(3))a(2-sqrt(3))(d)8a(2-sqrt(3))

Consider a branch of the hypebola x^2-2y^2-2sqrt2x-4sqrt2y-6=0 with vertex at the point A. Let B be one of the end points of its latus rectum. If C is the focus of the hyperbola nearest to the point A, then the area of the triangle ABC is (A) 1-sqrt(2/3) (B) sqrt(3/2) -1 (C) 1+sqrt(2/3) (D) sqrt(3/2)+1

The equation of the line that touches the curves y=x|x| and x^2+(y-2)^2=4 , where x!=0, is (a)y=4sqrt(5)x+20 (b)y=4sqrt(3)-12 (c)y=0 (d) y=-4sqrt(5)x-20