The equation in slope form of a normal to y^2=4ax is y=mx-2am-am^3 (learn this). Comparing it with y=mx+c we get condition for normality as c=-2am-am^3.For the given equation c=-4a root2 and m=root2.Putting given m in condition of normality we get -4root2a which is equal to c. therefore this equation is a normal. Now using the formula of the length of chord = 4/m^2{(1+m^2)*(a*(a-mc))}^1/2. And substituting a, m and c we get the required result.

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