Two parabolas `y^(2)=4a(x-lamda_(1))andx^(2)=4a(y-lamda_(2))` always touch each other (`lamda_(1),lamda_(2)` being variable parameters). Then their point of contact lies on a

Parabola y^2=4a(x-c_1) and x^2=4a(y-c_2) where c_1 and c_2 are variables, touch each other. Locus of their point of contact is

The matrix A={:[(lamda_(1)^(2),lamda_(1)lamda_(2),lamda_(1)lamda_(3)),(lamda_(2)lamda_(1),lamda_(2)^(2),lamda_(2)lamda_(3)),(lamda_(3)lamda_(1),lamda_(3)lamda_(2),lamda_(3)^(2))]:} is idempotent if lamda_(1)^(2)+lamda_(2)^(2)+lamda_(3)^(2)=k where lamda_(1),lamda_(2),lamda_(3) are non-zero real numbers. Then the value of (10+k)^(2) is . . .

Value of lamda so that point (lamda,lamda^(2)) lies between the lines |x+2y|=3 is

If (x+y)^(2)=2(x^(2)+y^(2))and(x-y+lamda)^(2)=4,lamdagt0, then lamda is equal to :

A variable parabola y^(2) = 4ax, a (where a ne -(1)/(4)) being the parameter, meets the curve y^(2) +x -y- 2 = 0 at two points. The locus of the point of intersecion of tangents at these points is

If the lines (x-4)/1=(y-2)/1=(z-lamda)/3 and x/1=(y+2)/2=z/4 intersect each other, then lamda lies in the interval

Find the locus of midpoint of family of chords lamdax+y=5(lamda is parameter) of the parabola x^(2)=20y

If equation (lamda^(2)-5lamda+6)x^(2)+(lamda^(2)-3lamda+2)x+(lamda^(2)-4)=0 is satisfied by more than two values of x , find the parameter lamda .

The point (alpha^(2)+2lamda+5,lamda^(2)+1) lies on the line x+y=10 for:

If alpha, beta are the roots fo the equation lamda(x^(2)-x)+x+5=0 . If lamda_(1) and lamda_(2) are two values of lamda for which the roots alpha, beta are related by (alpha)/(beta)+(beta)/(alpha)=4/5 find the value of (lamda_(1))/(lamda_(2))+(lamda_(2))/(lamda_(1))