The equation `(x-alpha)^2+(y-beta)^2=k(l x+m y+n)^2` represents (a) a parabola for `k = (l^2+m^2)^(-1)` (b) an ellipse for `0 < k < (l^2+m^2)^(-1)` (c)a hyperbola for `k >(1^2+m^2)^(-1)` (d) a point circle for `k=0`

The equation (x^2)/(1-k)-(y^2)/(1+k)=1, k gt 1 represents:-

The equation x^(2) - 2xy +y^(2) +3x +2 = 0 represents (a) a parabola (b) an ellipse (c) a hyperbola (d) a circle

The equation |sqrt(x^2+(y-1)^2)-sqrt(x^2+(y+1)^2)|=K will represent a hyperbola for (a) K in (0,2) (b) K in (-2,1) K in (1,oo) (d) K in (0,oo)

The equation, 2x^2+ 3y^2-8x-18y+35= K represents (a) no locus if k gt 0 (b) an ellipse if k lt 0 (c) a point if k = 0 (d) a hyperbola if k gt0

In X-Y plane, the path defined by the equation (1)/(x^(m))+(1)/(y^(m)) +(k)/((x+y)^(n)) =0 , is (a) a parabola if m = (1)/(2), k =- 1, n =0 (b) a hyperbola if m =1, k =- 1, n=0 (c) a pair of lines if m = k = n =1 (d) a pair of lines if m = k =- 1, n =1

If the equation |sqrt((x - 1)^(2) + y^(2) ) - sqrt((x + 1)^(2) + y^(2))| = k represents a hyperbola , then k belongs to the set

If (alpha, beta) is the foot of perpendicular from (x_1, y_1) to line lx+my+n=0 , then (A) (x_1 - alpha)/l = (y_1 - beta)/m (B) (x-1 - alpha)/l) = (lx_1 + my_1 + n)/(l^2+m^2) (C) (y_1 - beta)/m = (lx_1 + my_1 +n)/(l^2 +m^2) (D) (x-alpha)/l = (lalpha+mbeta+n)/(l^2+m^2)

If alpha=tan^(-1)((4x-4x^3)/(1-6x^2+x^2)),beta=2sin^(-1)((2x)/(1+x^2)) and tanpi/8=k , then (a) alpha+beta=pi for x in [(1,1)/k] (b) alpha+beta for x in (-k , k) (c) alpha+beta=pi for x in [(1,1)/k] (d) alpha+beta=0 for x in [-k , k]

If the line l x+m y+n=0 touches the parabola y^2=4a x , prove that ln=a m^2

If the line l x+m y+n=0 touches the parabola y^2=4a x , prove that ln=a m^2