UPSC NDA Math Model Paper 4

Given: The line y = x + 1 is a tangent to the curve $y^2 = 4x$ As we know that, the equation of tangent to the parabola $y^{2}=4 a x$ at the point $(x_{1}, y_{1})$ is given by : $y \cdot y_{1} = 2 a \cdot (x+x_{1})$ Let (1, -2) be the point of contact $\Rightarrow -2y = 4 \cdot (x + 1)$ $\Rightarrow y = -2 (x + 1)$ So, when (1, -2) is the point of contact the equation of tangent to the given curve is y = - 2 (x + 1) But as we know that, equation of the tangent to the given curve is y = x + 1 So, (1,- 2) is not the point at which the line y = x + 1 is a tangent to the curve $y^2 = 4x.$ Similarly, let us suppose (1, 2) be the point of contact $\Rightarrow 2y = 4 \cdot (x + 1)$ $\Rightarrow y = 2 \cdot (x + 1)$ So, when (1, 2) is the point of contact the equation of tangent to the given curve is $y = 2 (x + 1)$ But as we know that, equation of the tangent to the given curve is $y = x + 1$ So, (1, 2) is not the point at which the line y = x + 1 is a tangent to the curve $y^{} = 4x$ . Hence, option D is the correct answer.