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

The length of the chord of the parabola y^2=x which is bisected at the point (2, 1) is (a) 2sqrt(3) (b) 4sqrt(3) (c) 3sqrt(2) (d) 2sqrt(5)

If two different tangents of y^2=4x are the normals to x^2=4b y , then (a) |b|>1/(2sqrt(2)) (b) |b| 1/(sqrt(2)) (d) |b|<1/(sqrt(2))

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)

The two parabolas y^(2)=4x" and "x^(2)=4y intersect at a point P, whose abscissas is not zero, such that

Find the point at which the tangent to the curve y=sqrt(4x-3)-1 has its slope 2/3 .

An equilateral triangle S A B is inscribed in the parabola y^2=4a x having its focus at Sdot If chord A B 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))

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)

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)

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