Question

The equation of the tangent to the parabola, \[y^{2}=16x\text{ at}(1,4)\] is______.

Options

• \[x=4y\]

• \[2x+y+2=0\]

• \[y=4x\]

• \[2x-y+2=0\]

Answer 1

The equation of the tangent to the parabola, \[y^{2}=16x\text{ at}(1,4)\] is \[2x-y+2=0\].

Explanation:

The parabola is \[y^2=16x,\mathrm{so~}4a=16\Rightarrow a=4.\]

The standard equation of the tangent to \[y^2=4ax\] at a points \[(x_1,y_1)\] is:

                                   \[y\cdot y_1=2a(x+x_1)\]

At the point (1,4) :

                                   \[y\cdot4=2(4)(x+1)\]

                                  \[4y=8x+8\]

                                  \[8x-4y+8=0\]

                                  \[2x-y+2=0\]