Answer

Verified

373.8k+ views

**Hint:**In this problem we have given the equation of the tangent to a curve and asked to calculate the constant value in the curve equation which is $c$. We know that the slope of the tangent and the slope of the curve at tangent point are equal. But here we don’t have the tangent point. So, we will calculate the derivative of the given curve and equate it to the slope of the tangent. Here we will get an equation. Now we will observe all the equations we have and use proper substitutions to solve them. After solving the equations, we will get the coordinates of the tangent point. Now we will use the coordinates of the tangent point to get the value of $c$.

**Complete step by step solution:**

Given that, $y=\dfrac{3}{2}x+6$ is tangent to the curve $y=c\sqrt{x}$.

Considering the equation of the curve which is $y=c\sqrt{x}$.

Differentiating the above equation with respect to $x$, then we will get

$\begin{align}

& \Rightarrow \dfrac{dy}{dx}=\dfrac{d}{dx}\left( c\sqrt{x} \right) \\

& \Rightarrow \dfrac{dy}{dx}=c\dfrac{d}{dx}\left( \sqrt{x} \right) \\

\end{align}$

We have the differentiation formula $\dfrac{d}{dx}\left( \sqrt{x} \right)=\dfrac{1}{2\sqrt{x}}$. Substituting this value in the above equation, then we will get

$\Rightarrow \dfrac{dy}{dx}=\dfrac{c}{2\sqrt{x}}$

Considering the equation of the tangent which is $y=\dfrac{3}{2}x+6$.

The given tangent is in the form of slope intercept form i.e., $y=mx+c$, where $m$is the slope of the line. So, comparing the given tangent equation with $y=mx+c$, then we will get slope of the tangent as

$m=\dfrac{3}{2}$.

Equating the above slope value with the derivative of the curve, then we will get

$\begin{align}

& \Rightarrow \dfrac{c}{2\sqrt{x}}=\dfrac{3}{2} \\

& \Rightarrow c=3\sqrt{x} \\

\end{align}$

In this problem up to now we have the equations $c=3\sqrt{x}$, $y=\dfrac{3}{2}x+6$, $y=c\sqrt{x}$. By observing the all the above equations, we are going to substitute $c$ value in equation of curve and then equate it to equation of tangent, then we will get

$\Rightarrow 3\sqrt{x}\sqrt{x}=\dfrac{3}{2}x+6$

We know that $\sqrt{x}\sqrt{x}=x$, then we will have

$\Rightarrow 3x=\dfrac{3}{2}x+6$

Simplifying the above equation, then we will get

$\begin{align}

& \Rightarrow 3x=3\left( \dfrac{x}{2}+2 \right) \\

& \Rightarrow x=\dfrac{x}{2}+2 \\

& \Rightarrow x-\dfrac{x}{2}=2 \\

& \Rightarrow x=4 \\

\end{align}$

Substituting this value in the equation $c=3\sqrt{x}$, then we will get

$\begin{align}

& \Rightarrow c=3\sqrt{4} \\

& \Rightarrow c=3\times 2 \\

& \Rightarrow c=6 \\

\end{align}$

Hence the value of $c$ is $6$, from this the equation of the curve will be $y=6\sqrt{x}$. Now the graph of the curve and the tangent will be

**Note:**In this problem they have not asked to calculate the tangent point, so we have just calculated the $x$ coordinate of the point in the above solution. If they have asked to calculate the tangent point, then we will substitute the calculated $x$ value in either equation of the tangent or equation of curve.

Recently Updated Pages

Cryolite and fluorspar are mixed with Al2O3 during class 11 chemistry CBSE

Select the smallest atom A F B Cl C Br D I class 11 chemistry CBSE

The best reagent to convert pent 3 en 2 ol and pent class 11 chemistry CBSE

Reverse process of sublimation is aFusion bCondensation class 11 chemistry CBSE

The best and latest technique for isolation purification class 11 chemistry CBSE

Hydrochloric acid is a Strong acid b Weak acid c Strong class 11 chemistry CBSE

Trending doubts

Give 10 examples for herbs , shrubs , climbers , creepers

Difference Between Plant Cell and Animal Cell

Write a letter to the principal requesting him to grant class 10 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Name 10 Living and Non living things class 9 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

List some examples of Rabi and Kharif crops class 8 biology CBSE

Write the 6 fundamental rights of India and explain in detail