\[(A)\; 1 \quad (B)\; 2 \quad (C)\; 3 \quad (D)\;\frac{1}{2}\]

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- $\large\frac{d}{dx}$$(x^n)=nx^{n-1}$
- Equation of tangent $y=mx+1$

Step 1:

The equation of the curve is $y^2=4x$

Differentiating with respect to $x$

$2y\large\frac{dy}{dx}$$=4$

$\large\frac{dy}{dx}=\frac{4}{2y}$

$\large\frac{dy}{dx}=\frac{2}{y}$

Step 2:

Slope of the tangent $=\large\frac{2}{y}$$=m$-----(1)

$(x_1,y_1)$ lies on $y^2=4x$

$y_1^2=4x_1$-----(2)

Equation of tangent $(x_1,y_1)$

$y-y_1=m(x-x_1)$-----(3)

$y=mx-mx_1+y_1$

We are given the equation of tangent

$y=mx+1$------(4)

Step 3:

Comparing (3) and (4)

$y_1-mx_1=1$------(5)

From (1) & (2)

$m=\large\frac{2}{y_1}$

$x_1=\large\frac{y_1^2}{4}$

Step 4:

Put these value in equation (5)

$y_1-\large\frac{2}{y_1}.\frac{y_1^2}{4}$$=1$

$y_1-\large\frac{y_1}{2}$$=1$

$\large\frac{2y_1-y_1}{2}$$=1$

$\large\frac{y_1}{2}$$=1$

$\Rightarrow y_1=2$

$m=\large\frac{2}{y_1}=\frac{2}{2}$$=1$

Part (A) is the correct answer.

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