prove that the centres of the spheres which touch the lines y= mx , z= c and y= -mx , z= -c lie on the surface mxy + cz(1+m^2)=0
Answers
prove that the centres of the spheres which touch the lines y= mx , z= c and y= -mx , z= -c lie on the surface mxy + cz(1+m^2)=0
line Y=mX+c touches the parabola Y²=4aX if c=a/m .
Given:
equation of the line y = mx + c
equation of parabola , y² = 4ax
that line touches the parabola which means the line is a
tangent touching the parabola at only one point.
Hence, the points of intersection of the line with the parabola
should result in only 1 distinct root.
To Find:
The points of intersection of line y = mx + c with y² = 4ax
(mx + c)² = 4ax
m²x² + (2cm - 4a)x + c² = o
which is quadratic in x, but since the given line is a tangent
line touching at only one point, the above equation should have
equal roots, hence Discriminant = 0
SOLUTION:
(2mc - 4a)² = 4m²c²
By simplifying we get
4m²c² - 16amc + 16a² = 4m²c²
a² = amc
a(a - mc) = 0
Since a cannot be 0,
a - mc = 0
c = a/m
Hence, the line Y=mX+c touches the parabola Y²=4aX if c=a/m
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