Prove that the two parabolas `y^2=4ax` and `x^2=4by` intersects at an angle of `tan^1[(3a^(1/3)b^(1/3))/(2(a^(2/3)+b^(2/3)))]`

If the tangents to the parabola y^(2)=4ax at (x_(1),y_(1)),(x_(2),y_(2)) intersect at (x_(3),y_(3)) then

The parabolas y^(2)=4ac and x^(2)=4by intersect orthogonally at point P(x_(1),y_(1)) where x_(1),y_(1)!=0 provided (A)b=a^(2)(B)b=a^(3)(C)b^(3)=a^(2)(D) none of these

Prove that tan^(-1)""(3a^(2)x-x^(3))/(a^(3)-3ax^(2))=3tan^(-1)""x/a .

Prove that 2tan^(-1)((1)/(2))=tan^(-1)((4)/(3))

Angle between the parabolas y^(2)=4(x-1) and x^(2)+4(y-3)=0 at the common end of their locus rectum,is - (A) tan^(-1)(1)(B)tan^(-1)(1)+cot^(-1)(2)+cot^(-1)(3)(C)tan^(-1)(sqrt(3))(D)tan^(-1)(2)+tan^(-1)(3)

If a circle intersects the parabola y^(2) = 4ax at points A(at_(1)^(2), 2at_(1)), B(at_(2)^(2), 2at_(2)), C(at_(3)^(2), 2at_(3)), D(at_(4)^(2), 2at_(4)), then t_(1) + t_(2) + t_(3) + t_(4) is

A and B are two points on the parabola y^(2) = 4ax with vertex O. if OA is perpendicular to OB and they have lengths r_(1) and r_(2) respectively, then the valye of (r_(1)^(4//3)r_(2)^(4//3))/(r_(1)^(2//3)+r_(2)^(2//3)) is

The angle between the curves y^2=x and x^2=y at (1,\ 1) is tan^(-1)4/3 (b) tan^(-1)3/4 (c) 90o (d) 45o

Statement 1: The line ax+by+c=0 is a normal to the parabola y^(2)=4ax. Then the equation of the tangent at the foot of this normal is y=((b)/(a))x+((a^(2))/(b)). Statement 2: The equation of normal at any point P(at^(2),2at) to the parabola y^(2)= 4ax is y=-tx+2at+at^(3)