Question

Which of the following is (are) the coordinate(s) of the points on the curve $y=x^2+3x+4$, the tangents at which pass through the origin?

• A. (2, 14)
• B. (2, -14)
• C. (2, 2)
• D. (-2, 2)

Answer 1

Answer: (A), (D)

The correct options are
A (2, 14)
D (-2, 2)

Let P(h, k) be a point on the given curve such that the tangent at P passes through the origin. Since, P(h, k) lies on the given curve $y=x^2+3x+4$.
$\therefore k=h^2+3h+4....\left[1\right]$
The equation of the curve is $y=x^2+3x+4$
Differentiating w.r.t. x, we get
$\frac{\mathrm{dy}}{\mathrm{dx}}=2x+3\Rightarrow {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}_P=2h+3.$
The equation of the tangent at P(h, k) is
$y-k={\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}_P\left(x-h\right)\Rightarrow y-k=\left(2h+3\right)\left(x-h\right)\mathrm{It}\mathrm{passes}\mathrm{through}\mathrm{the}\mathrm{origin}i.e.\left(0,0\right).\therefore 0-k=\left(2h+3\right)\left(0-h\right)\Rightarrow k=2h^2+3h...\left(2\right]\mathrm{Subtracting}\left(2\right]\mathrm{from}\left(1\right],\mathrm{we}\mathrm{get}-h^2+4=0\Rightarrow h=\pm 2\mathrm{From}\left(2\right],\mathrm{If}h=2,\mathrm{then}y=14\mathrm{If}h=-2,\mathrm{then}y=2$
$\text{Hence, the required points are (2, 14) and (-2, 2).}$