If the straight line `(x-alpha)/(l)=(y-beta)/(m)=(z-gamma)/(n)` intersect the curve `ax^2+by^2=1, z=0,` then prove that `a(alphan-gammal)^2+b(betan-gammam)^2=n^2`

If the straight line xcosalpha+ysinalpha=p touches the curve (x^2)/(a^2)+(y^2)/(b^2)=1, then prove that a^2cos^2alpha+b^2sin^2alpha=p^2dot

If the straight line xcosalpha+ysinalpha=p touches the curve (x^2)/(a^2)-(y^2)/(b^2)=1, then prove that a^2cos^2alpha-b^2sin^2alpha=p^2dot

If the straight lines: (x-1)/k=(y-2)/2=(z-3)/3 and (x-2)/3 =(y-3)/k =(z-1)/2 intersect at a point, then the integer k is equal to :

If straight line lx + my + n=0 is a tangent of the ellipse x^2/a^2+y^2/b^2 = 1, then prove that a^2 l^2+ b^2 m^2 = n^2.

If the line l x+m y+n=0 touches the parabola y^2=4a x , prove that ln=a m^2

If the pair of straight lines Ax^(2)+2Hxy+By^(2)=0 be conjugate diameters of the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 , then prove that Aa^(2)=Bb^(2).

If 1, omega, omega^(2) are the three cube roots of unity and alpha, beta, gamma are the cube roots of p, plt0 , then for any x, y, z the expression (x alpha+y beta+z gamma)/(x beta+y gamma+z alpha)=

Let the line (x-2)/3=(y-1)/(-5)=(z+2)/2 lie in the plane x+3y-alpha z+beta =0 . Then (alpha,beta) equals :

If alpha,beta and gamma are roots of x^(3) -2x+1=0,then the value of Sigma((1)/(alpha+beta-gamma)) is