p(2ap,ap^2) is any point on the parabolax^2 = 4ay. The normal at p meets the y-axis at g. N is a point such that p is the midpoint of GN. Show that the equation of the locus of N is a parabola and find its vertex and focal length.

Question

p(2ap,ap^2) is any point on the parabolax^2 = 4ay. The normal at p meets the y-axis at g. N is a point such that p is the midpoint of GN. Show that the equation of the locus of N is a parabola and find its vertex and focal length.

Vikas TU · Answer

Dear Student,
slope of tangent to given parabola= at g is
m1=(dy/dx) at g= 2/8= ¼
also slope at N, m2= (dy/dx) at N= 1
as we found that y^2=4x
here a=1, therefore it is a parabola. [proved]
now,vertex=(0,0)
and focal length=a=1 units [ans]

Cheers!!

Regards,

Vikas (B. Tech. 4^th year
Thapar University)

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p(2ap,ap^2) is any point on the parabolax^2 = 4ay. The normal at p meets the y-axis at g. N is a point such that p is the midpoint of GN. Show that the equation of the locus of N is a parabola and find its vertex and focal length.

p(2ap,ap^2) is any point on the parabolax^2 = 4ay. The normal at p meets the y-axis at g. N is a point such that p is the midpoint of GN. Show that the equation of the locus of N is a parabola and find its vertex and focal length.

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p(2ap,ap^2) is any point on the parabolax^2 = 4ay. The normal at p meets the y-axis at g. N is a point such that p is the midpoint of GN. Show that the equation of the locus of N is a parabola and find its vertex and focal length.

p(2ap,ap^2) is any point on the parabolax^2 = 4ay. The normal at p meets the y-axis at g. N is a point such that p is the midpoint of GN. Show that the equation of the locus of N is a parabola and find its vertex and focal length.

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