# A rectangle #OABC# is drawn so that #O# is the vertex of parabola #y=x^2# and #OA# and #OB# are two chords drawn on the parabola. What is the locus of point #C#?

##
(A). #y=x^2+2#

(B). #y=-x^2+4#

(C). #y=2x^2#

(D). #y=-2x^2+4#

(A).

(B).

(C).

(D).

##### 2 Answers

(A)

#### Explanation:

The parabola

As the sides

Now intersection of

As

Below is shown the graph and rectangle relating to

graph{(y-x^2)(2y+x)(y-2x)(8x-4y+5)(2x+4y-20)(y-x^2-2)=0 [-4.84, 5.16, -0.38, 4.62]}

I think that name of the rectangle should be

Let the coordinates of

A

#->(t_1,t_1^2)#

B#->(t_2,t_2^2)#

C#->(h,k)#

O#->(0,0)#

Gradient of OA

Gradient of OB

As OA and OB are adjacent sides of the rectangle then product of their gradients should be

Hence

Now the diagonals ÂșC and

So coordinates of mid point of

And coordinates of mid point of

Hence

So

Converting