Question

Where does the line tangent to the graph y=x^2 at x= 2 intersect at x axis?

Answer 1

SOLUTION

TO DETERMINE

The point where the line tangent to the graph

$\sf{y = {x}^{2} }$ at x= 2 intersect at x axis

EVALUATION

Here the given equation of the curve is

$\sf{y = {x}^{2} } \: \: \: .....(1)$

Putting x = 2 we get y = 4

So the point is ( 2 , 4 )

Differentiating both sides of Equation (1) with respect to x we get

$\displaystyle \sf{ \frac{dy}{dx} = 2x }$

Now

$\displaystyle \sf{ \frac{dy}{dx} \bigg|_{(2,4)} = 4 }$

So the equation of the tangent at the point (2,4) is given by

$\sf{(y - 4) = 4(x - 2)}$

$\implies \sf{y - 4= 4x - 8}$

$\implies \sf{4x - y = 4}$

Now we rewrite the above equation of the line in intercept form

$\displaystyle \sf{ \frac{4x}{4} - \frac{y}{4} = 1}$

$\implies \displaystyle \sf{ \frac{x}{1} + \frac{y}{ - 4} = 1}$

∴ The line intersects x axis at ( 1,0) and y axis at (0,-4) respectively

Hence the line tangent to the graph

$\sf{y = {x}^{2} }$ at x= 2 intersect at x axis at (1,0)

FINAL ANSWER

Hence the line tangent to the graph

$\sf{y = {x}^{2} }$ at x= 2 intersect at x axis at (1,0)

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