1For a parabola at any point (at2, 2at) slope of normal is given by -t
=> m= -t
we can write the point as (am2, -2am)
if this point lies on the given line then :
-2am = am3+ c
give the necessary condition for a line y=mx + c to be normal to :
1)ellipse
2)parabola
3)hyperbola
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8 Answers
1
1For hyperbola, the equation of normal at any point(asec θ, btanθ) is-
axsinθ+ by= (a2+b2)tanθ
equation of the line is y-mx=c
These would represent the same line if
asinθ/-m = b = (a2+b2)tanθ/c
→ cosecθ=-a/mb and cotθ= a2+b2/bc
cosec2θ-cot2θ=1
Solving,
a2c2=m2[b2c2+(a2+b2)2]
1For ellipse,
any equation is ax secθ- by cosecθ=a2-b2
and line is y-mx=c
Solve it in the same way as in for hyperbola to get the result..