The number of points on the parabola y^2=x at which the slope of the normal drawn at the point is equal to the x-coordinate of that point is

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Correct answer: C. (c) Given, y^2=x Hence y^ '= 12y Let the point be (x_1 ,y_1 ). ∴y_1^2=x_1......(i) ∴ Slope of normal at (x_1 ,y_1 )= -1 (y^ ' )_ (x_1 ,y_1 ) = -1 ( 12y_1 )=-2y_1 It is given that, -2y_1=x_1 … … ( ii ) From Eqs. (i) and (ii), we have y_1^2=-2y_1 ⇒y_1^2+2y_1=0 ⇒y_1(y_2+2)=0 ⇒y_1=0 ,-2 ∴x_1=0 ,4 Required points are (0 ,0 ) , (4 ,-2 )

$The number of points on the parabola y^{2} = x at which the slope of the normal drawn at the point is equal to the x -coordinate of that point is$

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