Mathematics - Application of Derivative Question with Solution | TestHub

Every member of the family passes through the intersection of $2x-y=0$ and $2x+y-4=0$, i.e. (1, 2). Curve passes through $(9,0):0=3a+9b$; slope at $(9,0):\frac{a}{2\sqrt{9}}+b=-\frac{1}{2}$. Solving, $a=3,b=-1$, so $y=3\sqrt{x}-x$ Let the tangent at $\left(x_0,y_0\right)$ pass through $(1,2)$. Slope $m=\frac{3}{2\sqrt{x_0}}-1$ and tangent $y-y_0=m\left(x-x_0\right)$ passes (1, 2): $2-3\sqrt{x_0}+x_0=\left(\frac{3}{2\sqrt{x_0}}-1\right)\left(1-x_0\right)\Rightarrow \sqrt{x_0}=1\Rightarrow x_0=1,m=\frac{1}{2}$ Tangent: $y-2=\frac{1}{2}(x-1)\Rightarrow x-2y+3=0$