Question

Find the coordinates of the point on the curve y² = 3 − 4x where tangent is parallel to the line 2x + y − 2 = 0.

Answer 1

Let (x₁, y₁) be the required point.
Slope of the given line = $-$2

$$\begin{gathered} \mathrm{Since},\mathrm{the}\mathrm{point}\mathrm{lies}\mathrm{on}\mathrm{the}\mathrm{curve}. \\ \mathrm{Hence},{y_1}^2=3-4x_1...\left(1\right) \\ \mathrm{Now},y^2=3-4x \\ \Rightarrow 2y\frac{dy}{dx}=-4 \\ \therefore \frac{dy}{dx}=\frac{-4}{2y}=\frac{-2}{y} \\ \mathrm{S}\text{lope of the tangent =}{\left(\frac{dy}{dx}\right)}_{\left(x_1,y_1\right)}\text{=}\frac{-2}{y_1} \\ \text{Given:} \\ \text{Slope of the tangent = Slope of the line} \\ \Rightarrow \frac{-2}{y_1}=-2 \\ \Rightarrow y_1=1 \\ \text{From (1), we get} \\ 1=3-4x_1 \\ \Rightarrow -2=-4x_1 \\ \Rightarrow x_1=\frac{1}{2} \\ \therefore \left(x_1,y_1\right)=\left(\frac{1}{2},1\right) \end{gathered}$$