VJC P1 Q6

vjc p1q6

A curve has equation y^2=4x and a line l has equation 2x-y+1=0 as shown above.

B(b, 2\sqrt{b}) is a fixed point on C and A is an arbitrary point on l. State the geometrical relationship between the line segment AB and l is the distance from B to A is the least.

Taking the coordinates of A as (a, 2a+1), find an equation relating a and b for which AB is the least.

Deduce that when AB is the least, (AB)^2 = m (2b - 2\sqrt{b} +1)^2 where m is a constant to be found. Hence or otherwise, find the coordinates of the point on C that is nearest on l.

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