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 (x-1)/3=(y-2)/2=(z-3)/2 intersect the curve ax^(2) + by^(2) = 1, z = 0 find the value of 49a + 4b.

The line (x-2)/(3)=(y+1)/(2)=(z-1)/(-1) intersects the curve x^2+y^2=r^2, z=0 , then

The line (x-2)/(3)=(y+1)/(2)=(z-1)/(-1) intersects the curve xy=c^2, z=0, if c is equal to

If the line (x-1)/(5)=(y-3)/(2)=(z-3)/(2) intersects the curve x^(2)-y^(2)=k^(2), z=0 , then the value of 2k can be equal to

Prove that a line which passes through the point (alpha, beta, gamma) and intersects the curve y=0, z^2=4ax lies on the surface (bz-gammay)^2=4a(beta-y)(betax-alphay)

If the line l x+m y+n=0 touches the circle x^2+y^2=a^2 , then prove that (l^2+m^2)a^2=n^2dot

A line L passing through (1, 2, 3) and perpendicular to the line L_(1):(x-1)/(-2)=(y+1)/(3)=(z-5)/(4) is also intersecting the line L_(1) . If the line L intersects the plane 2x+y+z+6=0 at point (alpha, beta, gamma) , then the value of 2020alpha+beta+2gamma is equal to

A variable line ax+by+c=0 , where a, b, c are in A.P. is normal to a circle (x-alpha)^2+(y-beta)^2=gamma , which is orthogonal to circle x^2+y^2-4x-4y-1=0 .The value of alpha + beta+ gamma is equal to

If alpha,beta are roots of x^2+p x+1=0a n dgamma,delta are the roots of x^2+q x+1=0 , then prove that q^2-p^2=(alpha-gamma)(beta-gamma)(alpha+delta)(beta+delta) .

If the lines (x-3)/(2)=(y-5)/(2)=(z-4)/(lambda) and (x-2)/(lambda)=(y-6)/(4)=(z-5)/(2) intersect at a point (alpha, beta, gamma) , then the greatest value of lambda is equal to