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# Help me understand! Let O be the vertex and Q be any point on the parabola, x2 = 8y. If the point Pdivides the line segment OQ internally in the ratio 1 : 3, then the locus of P is :

Let O be the vertex and Q be any point on the parabola, x2 = 8y. If the point P

divides the line segment OQ internally in the ratio 1 : 3, then the locus of P is :

• Option 1)

x2 = y

• Option 2)

y=  x

• Option 3)

y=  2x

• Option 4)

x2 =  2y

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As learnt in

Locus -

Path followed by a point p(x,y) under given condition (s).

- wherein

It is satisfied by all the points (x,y) on the locus.

Parametric coordinates of parabola -

$x= at^{2}$

$y= 2at$

- wherein

For the parabola.

$y^{2}=4ax$

Selection formula -

$x= \frac{mx_{2}+nx_{1}}{m+n}$

$y= \frac{my_{2}+ny_{1}}{m+n}$

- wherein

If P(x,y) divides the line joining A(x1,y1) and B(x2,y2) in ration $m:n$

Parametric coordinates $(at^{2},2at)$

By section formula $h\:=\frac{4t}{4}\:=t$

$P\:=\frac{2t^{2}}{4}\:=\frac{t^{2}}{2}$

eliminating t, we get h2 = 2k

=> x2 = 2y

Option 1)

x2 = y

This option is incorrect.

Option 2)

y=  x

This option is incorrect.

Option 3)

y=  2x

This option is incorrect.

Option 4)

x2 =  2y

This option is correct.

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