Two perpendicular tangents to y^2=4 ax where a>0 always intersect on the line (a) x+a=0 (b) x+4

step 1 So in this question, two perpendicular tangents to this parabola always intersect on the line. T

Two perpendicular tangents to y^2=4 ax where a>0 always...

Key Concepts

Intersection of Tangent Lines

The point where two tangents intersect is found by solving their equations simultaneously. In problems involving perpendicular tangents, the intersection point can be expressed in a form that reveals its location relative to the parabola’s standard elements, such as symmetry lines or axes, leading to a more profound geometrical insight into the parabola.

Perpendicular Tangents

Perpendicular tangents are pairs of tangent lines that intersect at a right angle (90°). The condition for perpendicularity, typically the product of their slopes being -1, is used to establish relationships between the parameters or points of tangency. This concept is important in determining the locus of the intersection point of such tangents.

Tangent to a Parabola

A tangent to a parabola is a line that touches the curve at exactly one point without intersecting it elsewhere. The analytical expression of the tangent line can be derived using the point of contact and the slope, and its properties help in solving problems related to contact points and intersection with other lines.

Parabola Geometry

Parabolas are conic sections defined by a specific quadratic relation between the y and x coordinates. Their geometric properties, such as focus, directrix, and axis of symmetry, play a crucial role in analyzing their tangents and normals, which are fundamental in conic studies.

Parametric Representation

The parametric representation of a parabola provides a convenient way to express the coordinates of any point on the curve in terms of a parameter. This form is essential for deriving the equation of the tangent line in a simplified manner and also for handling problems involving tangents drawn from points on the curve.

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