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Question

Let ' O be the origin and A be a point on the curve y to the power of 2 end exponent equals 4 x then locus of the midpoint of OA is

  1. x to the power of 2 end exponent equals 4 y    
  2. x to the power of 2 end exponent equals 2 y    
  3. y to the power of 2 end exponent equals 16 x    
  4. y to the power of 2 end exponent equals 2 x    

Hint:

Use Midpoint formula :
left parenthesis x subscript m space comma space y subscript m right parenthesis space equals space open parentheses fraction numerator x subscript 1 space plus x subscript 2 over denominator 2 end fraction comma space fraction numerator y subscript 1 space plus space y subscript 2 over denominator 2 end fraction close parentheses

w h e r e space comma left parenthesis space x subscript m space comma space y subscript m right parenthesis space equals space c o o r d i n a t e s space o f space t h e space m i d p o i n t
left parenthesis space x subscript 1 space comma space y subscript 1 right parenthesis space equals space c o o r d i n a t e s space o f space t h e space f i r s t space p o i n t
left parenthesis space x subscript 1 space comma space y subscript 1 right parenthesis space equals space c o o r d i n a t e s space o f space t h e space s e c o n d space p o i n t

The correct answer is: y to the power of 2 end exponent equals 2 x


     Using Midpoint formula :
    left parenthesis x subscript m space comma space y subscript m right parenthesis space equals space open parentheses fraction numerator x subscript 1 space plus x subscript 2 over denominator 2 end fraction comma space fraction numerator y subscript 1 space plus space y subscript 2 over denominator 2 end fraction close parentheses

w h e r e space comma left parenthesis space x subscript m space comma space y subscript m right parenthesis space equals space c o o r d i n a t e s space o f space t h e space m i d p o i n t
left parenthesis space x subscript 1 space comma space y subscript 1 right parenthesis space equals space c o o r d i n a t e s space o f space t h e space f i r s t space p o i n t
left parenthesis space x subscript 1 space comma space y subscript 1 right parenthesis space equals space c o o r d i n a t e s space o f space t h e space s e c o n d space p o i n t

    Let Origin - O (0,0) and A (x,4) be a point on the curve y to the power of 2 end exponent equals 4 x
    Let left parenthesis space x subscript 1 space comma space y subscript 1 right parenthesis space space equals space open parentheses fraction numerator x plus space 0 space over denominator 2 end fraction comma space fraction numerator y space plus space 0 over denominator 2 end fraction close parentheses
    rightwards double arrow space x subscript 1 space equals space x over 2 space a n d space y subscript 1 space equals space y over 2
rightwards double arrow x space equals space 2 x subscript 1 space a n d space y space equals space 2 y subscript 1 space
    Substitute these values in y to the power of 2 end exponent equals 4 x
    rightwards double arrow open parentheses 2 y subscript 1 close parentheses squared equals 4 open parentheses 2 x subscript 1 close parentheses
rightwards double arrow 4 y subscript 1 squared equals 4 open parentheses 2 x subscript 1 close parentheses
rightwards double arrow y subscript 1 squared equals open parentheses 2 x subscript 1 close parentheses

T h u s comma space l o c u s space o f space m i d p o i n t space o f space O A space i s space space y squared space equals 2 x

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