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Section Formula

If O is the o...

Question

$If O is the origin and Q is a variable point on y^{2} = x . Fin the locus of the mid-point of OQ.$

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Solution

$Let P(h,k)be the point on the locus and let the coordinates of a are (a,b).Then, h = \frac{a + 0}{2} a n d \frac{b + 0}{2} = k [∵ p i s t h e m i d - p o i n t o f Q a n d t h e o r i g i n o] h = \frac{a}{2} a n d b = 2 k \Rightarrow a = 2 h a n d b = 2 k P o i n t Q l i e s o n t h e y^{2} = x, T h e n b^{2} = a [∵ Q : (a, b)] \Rightarrow (2 k)^{2} = 2 h [∵ a = 2 h a n d b = 2 k] \Rightarrow 4 k^{2} = 2 h \Rightarrow 2 k^{2} = h H e n c e, t h e l o c u s o f (h, k) i s 2 y^{2} = x .$

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