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

Let (h,k) be the point at which tangent is to be taken.

Given line is y = x+1 and curve is y² = 4x

We know that slope of tangent is .

For y² = 4x

Since tangent to be taken from (h,k).

So slope of tangent at (h,k) is :

For line y = x + 1

Comparing with y = mx + c

Slope is m.

….. (1)

Now,

Point (h,k) lie on the curve y² = 4x

⇒ (h,k) will satisfy the equation of curve.

Putting x = h, y = k in equation we get,

⇒ k² = 4h ….. (2)

Also,

Point (h,k) lie on the tangent.

⇒ (h,k) will satisfy the equation of line.

Putting x = h, y = k in equation we get,

⇒ k= mh + 1

Putting value of m from (1) we get,

K² = 2h + k

K² – k = 2h …. (3)

From (2),

k² = 4h

⇒ k² = 2(2h)

⇒ k² = 2(k² - k)

⇒ k² = 2k² - 2k

⇒ k² - 2k = 0

⇒ k(k-2) = 0

⇒ k = 0 and k = 2

When k = 0,

m = ∞

which cannot be possible

When k=2,

m = 1

So, the value of m is 1.