Question

The slope of a tangent drawn from the point (1, 1) to the parabola $y^2=4(x-y)$ is

• A. 1
• B. $-1$
• C. $\frac{1}{2}$
• D. $-\frac{1}{2}$

Answer 1

Answer: (A), (C)

The correct options are
A 1
C $\frac{1}{2}$

Let $m$ be the slope of the tangent from $(1,1)$.

The line $y-1=m(x-1)$ meets the parabola at two coincident points.

Eliminating $y$ between the line and the parabola,

$[1+m(x-1){]}^2-4x+4[1+m(x-1)]=0$

$\Rightarrow m^2(x-1{)}^2+(6m-4\left)\right(x-1)+1=0$

It is quadratic in $(x-1)$ with equal roots.

So, discriminant = 0
$\therefore (6m-4{)}^2-4m^2=0$

$\Rightarrow (6m-4-2m)(6m-4+2m)=0$
$\Rightarrow (m-1)(8m-4)=0$
$\Rightarrow m=1,\frac{1}{2}$