Question

Y^2=8X a tangent is passing through this parabola then what is the tangent

Answer 1

Y = mX + 2/m

Answer 2

The given parabola is $Y^2 = 8X$.

Comparing this with the standard form of a parabola $y^2 = 4ax$, we find that $4a = 8$, which implies $a = 2$.

There are two common forms for the general equation of a tangent to a parabola:

1. Tangent in terms of its slope ($m$):

   The equation of a tangent to the parabola $y^2 = 4ax$ with slope $m$ is given by:

   $y = mx + \frac{a}{m}$

   Substituting $a = 2$ and using variables $X$ and $Y$:

   $Y = mX + \frac{2}{m}$

2. Tangent at a point $(X_1, Y_1)$ on the parabola:

   The equation of the tangent to the parabola $Y^2 = 8X$ at a point $(X_1, Y_1)$ lying on the parabola is found by replacing $Y^2$ with $Y Y_1$ and $X$ with $\frac{X+X_1}{2}$:

   $Y Y_1 = 8 \left(\frac{X+X_1}{2}\right)$

   $Y Y_1 = 4(X+X_1)$

Given the general nature of the question "what is the tangent", the equation in terms of its slope is generally expected.