Question

# The line y = mx + 1 is a tangent to the curve y 2 = 4 x if the value of m is (A) 1 (B) 2 (C) 3 (D)

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Solution

## The given equation of the line is y=mx+1 and the equation of the curve is y 2 =4x . Solving the equation of line and curve to find the point of intersection. Put the value of y from equation of line in the equation of curve. ( mx+1 ) 2 =4x m 2 x 2 +1+2mx=4x m 2 x 2 −4x+2mx+1=0 m 2 x 2 +x( 2m−4 )+1=0 It is given that the line is tangent to the curve, so the line must touch the curve at one point only. Thus, the equation must have same roots and in that case the value of discriminant of the equation should be zero. ( 2m−4 ) 2 −4( m 2 )( 1 )=0 4 m 2 +16−16m−4 m 2 =0 16−16m=0 m=1 The value of m is 1. Thus, the correct option is (A).

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