JEE Main 2019 (Online) 12th January Evening ShiftQuestion 1
The tangent to the curve $y=x^{2}-5x+5$, parallel to the line $2y=4x+1$, also passes through the point :
Differentiation and its Application
Solution
Concept. Find the point where $y'=2$, write the tangent, and test the options. Formulas used. (1) Tangent at $(x_1,y_1)$: $y-y_1=m(x-x_1)$Step 1 - Point of contact. $2y=4x+1$ has slope $2$. $y'=2x-5=2$ gives $x=\frac72$ and $y=\frac{49}{4}-\frac{35}{2}+5=-\frac14$. Step 2 - Tangent. By (1): $y+\frac14=2\left(x-\frac72\right)$, i.e. $y=2x-\frac{29}{4}$. Step 3 - Check. At $x=\frac18$: $y=\frac14-\frac{29}{4}=-7$. So $\left(\frac18,-7\right)$ is on it. Answer: (D)