If the line y = mx + 1 is tangent

Question:

If the line y = mx + 1 is tangent to the parabola y2 = 4x then find the value of m.

Solution:

Given equations are,

y = mx + 1 & y2 = 4x

By solving given equations we get

(mx + 1)2 = 4x

Expanding the above equation we get

m2x2 + 2mx + 1 = 4x

On rearranging we get

m2x2 + 2mx – 4x + 1 = 0

m+x2 + x (2m – 4) + 1 = 0

As the line touches the parabola, above equation must have equal roots,

Discriminant (D) = 0

(2m – 4)2 – 4 (m2) (1) = 0

4m2 – 16m + 16 – 4m2 = 0

-16 m + 16 = 0

– m + 1 = 0

m = 1

Hence, the required value of m is 1.

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