If `(5,12)a n d(24 ,7)` are the foci of a hyperbola passing through the origin, then (a)`e=(sqrt(386))/(12)` (b) `e=(sqrt(386))/(13)` (c)`L R=(121)/6` (d) `L R=(121)/3`

If (5, 12) and (24, 7) are the foci of an ellipse passing through the origin, then find the eccentricity of the ellipse.

An ellipse having foci at (3, 3) and (-4,4) and passing through the origin has eccentricity equal to (a) 3/7 (b) 2/7 (c) 5/7 (d) 3/5

The vertices of a triangle are (0,0), (x ,cosx), and (sin^3x ,0),w h e r e0ltxltpi/2 the maximum area for such a triangle in sq. units is (a) (3sqrt(3))/(32) (b) (sqrt(3)/32) (c) 4/32 (d) (6sqrt(3)) /(32)

If the distance between the foci and the distance between the two directricies of the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 are in the ratio 3:2, then b : a is (a) 1:sqrt(2) (b) sqrt(3):sqrt(2) (c) 1:2 (d) 2:1

The eccentricity of the conjugate hyperbola of the hyperbola x^2-3y^2=1 is (a) 2 (b) 2sqrt(3) (c) 4 (d) 4/5

The vertices of a parallelogram A B C D are A(3,1),B(13 ,6),C(13 ,21), and D(3,16)dot If a line passing through the origin divides the parallelogram into two congruent parts, then the slope of the line is (a) (11)/(12) (b) (11)/8 (c) (25)/8 (d) (13)/8

One angle of an isosceles triangle is 120^0 and the radius of its incircle is sqrt(3)dot Then the area of the triangle in sq. units is (a) 7+12sqrt(3) (b) 12-7sqrt(3) (c) 12+7sqrt(3) (d) 4pi

In triangle A B C ,l e tR=c i r c u m r a d i u s ,r= i n r a d i u sdot If r is the distance between the circumcenter and the incenter, the ratio R/r is equal to (a) sqrt(2)-1 (b) sqrt(3)-1 (c) sqrt(2)+1 (d) sqrt(3)+1

If a/sqrt(b c)-2=sqrt(b/c)+sqrt(c/b), where a , b , c >0, then the family of lines sqrt(a)x+sqrt(b)y+sqrt(c)=0 passes though the fixed point given by (a) (1,1) (b) (1,-2) (c) (-1,2) (d) (-1,1)

A circle passes through the points A(1,0) and B(5,0), and touches the y-axis at C(0,h)dot . If /_A C B is maximum, then (a) h=3sqrt(5) (b) h=2sqrt(5) (c) h=sqrt(5) (d) h=2sqrt(10)