Answer
$$6=C$$
Work Step by Step
Differentiate with respect to \(x\)
\[
c \sqrt{x}=y
\]
\[
\frac{c}{2 \sqrt{x}}=y^{\prime}
\]
Let’s assume that the tangent occurs at \(x=a\)
since the slope of the line is \(\frac{3}{2}, y^{\prime}\) must also be \(\frac{3}{2}\)
\[
\frac{c}{2 \sqrt{a}}=\frac{3}{2}
\]
Multiply both sides by \(2 \sqrt{a}\)
\[
c=3 \sqrt{a}
\]
The line will meet the curve if and only if the following equation is true
\[
6+\frac{3}{2} a=c \sqrt{a}
\]
Substitute \(c\) = \(3 \sqrt{a}\)
\[
\begin{array}{c}
3 \sqrt{a} \cdot \sqrt{a}=\frac{3}{2} a+6 \\
6+\frac{3}{2} a=3 a
\end{array}
\]
Subtract \(\frac{3}{2} a\) from each sides
\[
6=\frac{3}{2} a
\]
\[
4=a
\]
Substitute this in \(3 \sqrt{a}=c,\) to have
\[
\begin{array}{c}
C=3 \sqrt{4} \\
C=6
\end{array}
\
