The equation of a tangent to the parabola y^2=8 x is y=x+2 ....

The equation of a tangent to the parabola $y^{2}=8 x$ is $y=x+2 .$ The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is $|2007|$
(A) $(-1,1)$
(B) $(0,2)$
(C) $(2,4)$
(D) $(-2,0)$

Key Concepts

Tangents from an External Point

When tangents are drawn from a point outside a conic section, such as a parabola, there are generally two possible tangents. Determining these tangents involves using the condition for tangency, which typically requires setting the discriminant of the resulting quadratic equation to zero. This process is often combined with additional conditions, such as perpendicularity, to uniquely identify the tangent lines or relevant points.

Perpendicular Lines

Two lines are perpendicular if the product of their slopes is -1. This property is essential in problems where one needs to find another line (or tangent) that is perpendicular to a known line, as it provides a direct relationship between their slopes and helps determine the equation of the unknown line.

Parabola

A parabola is a conic section defined as the set of all points that are equidistant from a fixed point called the focus and a fixed line called the directrix. Its geometric properties, such as symmetry and the relationship between its algebraic equation and its geometric orientation, are fundamental to understanding problems involving tangents and normals.

Tangent Line

A tangent line to a curve is a straight line that touches the curve at exactly one point and has the same slope as the curve at that point. For conic sections like parabolas, the equation of the tangent can be derived using methods such as implicit differentiation or by employing standard tangent formulas, and it plays a key role in analyzing the interaction between the line and the curve.

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