Question

If the line $y=mx+1$ is tangent to the parabola $y^2=4x$ then find the value of $m$.

Answer 1

Given: Line $y=mx+1$ …(i) is tangent of parabola $y^2=4x$ …(ii).

From (i), we get $x=\frac{y-1}{m}$ …(iii)

By solving (ii) and (iii),

$y^2=4\left(\frac{y-1}{m}\right)$

$\Rightarrow m{y}^2-4y+4=0$ …(iv)

So, If line (i) is tangent of parabola (ii), then (iv) should have only one solution.

i.e., both roots of the quadratic solution should be equal.

If $a{x}^2+bx+c$ has equal roots, then discriminant $b^2-4ac=0$

In (iv), $a=m,b=-4,c=4$

$\Rightarrow (-4{)}^2=4\cdot m\cdot 4$

$\Rightarrow 16=16m$

$\Rightarrow m=1$

Hence, the value of $m$ is $1$.