`x^2/(r^2-r-6)+y^2/(r^2-6r+5)=1` will represent the ellipse if `r` lies in the interval

The equation (x^2)/(2 - r) + (y^2)/(r - 5) + 1 = 0 represent an ellipse iff

The equation (x^2)/(1-r)-(y^2)/(1+r)=1,r >1, represents (a)an ellipse (b) a hyperbola (c)a circle (d) none of these

Let ABC be an equilateral triangle inscribed in the circle x^2+y^2=a^2 . Suppose pendiculars from A, B, C to the ellipse x^2/a^2+y^2/b^2=1,(a > b) meets the ellipse respectivelily at P, Q, R so that P, Q , R lies on same side of major axis as A, B, C respectively. Prove that the normals to the ellipse drawn at the points P Q nad R are concurrent.

If (y^2-5y+3)(x ^2+x+1)<2x for all x in R , then find the interval in which y lies.

If from a point P , tangents P Qa n dP R are drawn to the ellipse (x^2)/2+y^2=1 so that the equation of Q R is x+3y=1, then find the coordinates of Pdot

If a,b and c also represent the sides of a triangle and a,b,c are in g.p then the complete set of alpha^2 = (r^2 +r +1)/(r^2-r+1) is

If Sigma_(r=1)^(50) (r^2)/(r^2+(11-r)^2) , then the value of n is __________

Let S={(x,y) in R^(2):(y^(2))/(1+r)-(x^(2))/(1-r)=1} , where r ne pm 1 . Then S represents:

If the two circles (x+1)^2+(y-3)=r^2 and x^2+y^2-8x+2y+8=0 intersect in two distinct point,then (A) r > 2 (B) 2 < r < 8 (C) r < 2 (D) r=2