If `(a x^2+c)y+(a^(prime)x^2+c^(prime) )=0` and `x` is a rational function of `y` and `ac` is negative, then
a. `a c^(prime)+c^(prime)c=0`
b. `a//a '=c//c '`
c. `a^2+c^2=a^('2)+c^('2)`
d. `a a^(prime)+c c^(prime)=1`

If y is a function of xa n dlog(x+y)-2x y=0, then the value of y^(prime)(0) is (a)1 (b) -1 (c) 2 (d) 0

If the circles x^2+y^2+2a^(prime)x+2b^(prime)y+c^(prime)=0 and 2x^2+2y^2+2a x+2b y+c=0 intersect othrogonally, then prove that a a^(prime) + b b prime=c+c^(prime)/2dot

If a >0 and discriminant of a x^2+2b x+c is negative, then =a b a x+bb c b x+c a x+bb x+c0 is +v e b. (a c-b)^2(a x^2+2b x+c) c. -v e d. 0

If the normal to the given hyperbola at the point (c t , c/t) meets the curve again at (c t^(prime), c/t^(prime)), then (A) t^3t^(prime)=1 (B) t^3t^(prime)=-1 (C) t t^(prime)=1 (D) t t^(prime)=-1

The condition that one of the straight lines given by the equation a x^2+2h x y+b y^2=0 may coincide with one of those given by the equation a^(prime)x^2+2h^(prime)x y+b^(prime)y^2=0 is (a b^(prime)-a^(prime)b)^2=4(h a^(prime)-h^(prime)a)(b h^(prime)-b^(prime)h) (a b^(prime)-a^(prime)b)^2=(h a^(prime)-h^(prime)a)(b h^(prime)-b^(prime)h) (h a^(prime)-h^(prime)a)^2=4(a b^(prime)-a^(prime)b)(b h^(prime)-b^(prime)h) (b h^(prime)-b^(prime)h)^2=4(a b^(prime)-a^(prime)b)(h a^(prime)-h^(prime)a)

If the quadrilateral formed by the lines a x+b y+c=0,a^(prime)x+b^(prime)y+c=0,a x+b y+c^(prime)=0,a^(prime)x+b^(prime)y+c^(prime)=0 has perpendicular diagonals, then (a) b^2+c^2=b^('2)+c^('2) (b) c^2+a^2=c^('2)+a^('2) (c) a^2+b^2=a^('2)+b^('2) (d) none of these

Find the condition if lines x=a y+b ,z=c y+da n dx=a^(prime)y+b^(prime), z=c^(prime)y+d ' are perpendicular.

Let y=f(x) be a parabola, having its axis parallel to the y-axis, which is touched by the line y=x at x=1. Then, (a) 2f(0)=1-f^(prime)(0) (b) f(0)+f^(prime)(0)+f^(0)=1 (c) f^(prime)(1)=1 (d) f^(prime)(0)=f^(prime)(1)

If a >0 and discriminant of a x^2+2b x+c is negative, then |[a,b,ax+b],[b,c,bx+c],[ax+b,bx+c,0]| is +v e b. (a c-b)^2(a x^2+2b x+c) c. -v e d. 0

(log)_4 18 is (a) a rational number (b) an irrational number (c) a prime number (d) none of these