Find the coordinates of the points on the curve `y=x^2+3x+4` , the tangents at which pass through the origin.

Answer

Required points are `(2,14)` and `(-2,2)`

Explanation

Let us suppose that tangent is made at point `(x_{1}, y_{1})`
`y_{1}=x_{1}^{2}+3 x_{1}+4 \ldots(1)`


`m = dy / dx = 2 x+3`


`m` at `(x_{1}, y_{1})=2 x_{1}+3`


Equation of tangent= `y_{1}=(2 x_{1} + 3 ) x_{1} \ldots(2)`


On comparing eq(1) and eq(2)
$$
x_{1}^{2}+3 x_{1}+4=\left(2 x_{1}+3\right) x_{1}
...