Question

Find the equation of the tangent line to the curve y = x² − 2x + 7 which is (i) parallel to the line 2x − y + 9 = 0 (ii) perpendicular to the line 5y − 15x = 13.

Answer 1

(i) Slope of the given line is 2
$$\begin{gathered} \text{Let}\left(x_1,y_1\right)\text{be the point where the tangent is drawn to the curve}y=x^2-2x+7 \\ \mathrm{Since},\mathrm{the}\mathrm{point}\mathrm{lies}\mathrm{on}\mathrm{the}\mathrm{curve}. \\ \mathrm{Hence},y_1={x_1}^2-2x_1+7...\left(1\right) \\ \mathrm{Now},y=x^2-2x+7 \\ \Rightarrow \frac{dy}{dx}=2x-2 \\ \text{Slope of tangent at point}\left(x_1,y_1\right)=2x_1-2 \\ \text{Given that} \\ \text{Slope of tangent= Slope of the given line} \\ \Rightarrow 2x_1-2=2 \\ \Rightarrow 2x_1=4 \\ \Rightarrow x_1=2 \\ \mathrm{Now},y_1=4-4+7=7 \\ \therefore \left(x_1,y_1\right)=\left(2,7\right) \\ \text{Equation of tangent is,} \\ y-y_1=m\left(x-x_1\right) \\ \Rightarrow y-7=2\left(x-2\right) \\ \Rightarrow y-7=2x-4 \\ \Rightarrow 2x-y+3=0 \end{gathered}$$

(ii) Slope of the given line is 3
Slope of the line perpendicular to this line = $\frac{-1}{3}$

$$\begin{gathered} \text{Let}\left(x_1,y_1\right)\text{be the point where the tangent is drawn to the curve.} \\ \mathrm{Since},\mathrm{the}\mathrm{point}\mathrm{lies}\mathrm{on}\mathrm{the}\mathrm{curve}. \\ \mathrm{Hence},y_1={x_1}^2-2x_1+7...\left(1\right) \\ \mathrm{Now},y=x^2-2x+7 \\ \Rightarrow \frac{dy}{dx}=2x-2 \\ \text{Slope of tangent at}\left(x_1,y_1\right)\text{=}2x_1-2 \\ \text{Given that} \\ \text{Slope of tangent at}\left(x_1,y_1\right)\text{= Slope of the perpendicular line} \\ \Rightarrow 2x_1-2=\frac{-1}{3} \\ \Rightarrow 6x_1-6=-1 \\ \Rightarrow 6x_1=5 \\ \Rightarrow x_1=\frac{5}{6} \\ \mathrm{Now},y_1=\frac{25}{36}-\frac{10}{6}+7=\frac{25-60+252}{36}=\frac{217}{36} \\ \therefore \left(x_1,y_1\right)=\left(\frac{5}{6},\frac{217}{36}\right) \\ \text{Equation of tangent is,} \\ y-\frac{217}{36}=\frac{-1}{3}\left(x-\frac{5}{6}\right) \\ \Rightarrow \frac{36y-217}{36}=\frac{-6x+5}{18} \\ \Rightarrow 36y-217=-12x+10 \\ \Rightarrow 12x+36y-227=0 \end{gathered}$$