Dilating the Parabola: Exploring the Locus of the Midpoint of OP

The problem at hand involves a point P moving along the curve defined by y2 x4. We are interested in determining the locus of the midpoint M of OP, where O is the origin.

Understanding the Midpoint and the Curve

To begin, we define the point P as x1, y1, and since P lies on the curve, we have y1 ±sqrt(x14). This means P can be expressed as: [ P (x1, pmsqrt{x14}) ]

Locating the Midpoint

The midpoint M of OP is: [ M left(frac{x1}{2}, pmfrac{sqrt{x14}}{2}right) ] After simplification, this yields: [ M left(frac{x1}{2}, pmfrac{x12}{2}right) ]

Determining the Locus of the Midpoint

Since we have defined the relationship between M and P, we can now express M in terms of x and y coordinates: [ y pmfrac{x^2}{2} ] Squaring both sides to express in standard form: [ y^2 frac{x^2}{4} ] Multiplying both sides by 4: [ 4y^2 x^2 ] Rearranging, we get: [ x^2 - 4y^2 0 ] Or: [ x^2 4y^2 ] Dividing through by 2: [ frac{x^2}{4} y^2 ] We can express this as: [ y^2 frac{x^2}{4} ] Which simplifies to the standard form of a hyperbola: [ x^2 - 4y^2 0 ]

Alternative Method: Direct Substitution

Alternatively, we can directly substitute the midpoint coordinates x1 / 2 and y1 / 2 into the original curve equation: [ y^2 x^4 ] Replacing x and y: [ left(frac{y}{2}right)^2 left(frac{x}{2}right)^4 ] Simplifying: [ frac{y^2}{4} frac{x^4}{16} ] Multiply both sides by 16: [ 4y^2 4x^4 ] Divide by 4: [ y^2 frac{x^4}{4} ] Or: [ y^2 frac{x^2}{2} ] Which leads to the same standard form: [ x^2 - 4y^2 0 ]

Conclusion

The locus of the midpoint M of OP is defined by the equation: [ x^2 - 4y^2 0 ] This represents a hyperbola centered at the origin, dilated by a factor of 1/2.