The normals at two points P and Q on the parabola y^2=4 ax meet at a point R(x1, y1) on the para

step 1 In this problem of conic section, we have given that the normals at two points which is point P

The normals at two points P and Q on the parabola y^2=4 ax...

The normals at two points $\mathrm{P}$ and $\mathrm{Q}$ on the parabola $\mathrm{y}^{2}=4 \mathrm{ax}$ meet at a point $\mathrm{R}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$ on the parabola then $\mathrm{PQ}^{2}$ is (a) $\left(x_{1}-8 a\right)^{2}$ (b) $\left(x_{1}-8 a\right)\left(x_{1}+4 a\right)$ (c) $\left(x_{1}-4 a\right)\left(x_{1}+8 a\right)$ (d) $x_{1}^{2}-16 a^{2}$

The normals at two points $\mathrm{P}$ and $\mathrm{Q}$ on the parabola $\mathrm{y}^{2}=4 \mathrm{ax}$ meet at a point $\mathrm{R}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$ on the parabola then $\mathrm{PQ}^{2}$ is
(a) $\left(x_{1}-8 a\right)^{2}$
(b) $\left(x_{1}-8 a\right)\left(x_{1}+4 a\right)$
(c) $\left(x_{1}-4 a\right)\left(x_{1}+8 a\right)$
(d) $x_{1}^{2}-16 a^{2}$

Key Concepts

Distance Between Points on a Curve

Calculating the chord length, or the distance between two points on a curve, is a common application of the distance formula in coordinate geometry. In problems involving parabolas and their normals, determining the distance squared between two points can lead to algebraic relationships that tie together the coordinates of the intersection point of normals with the geometric parameters of the parabola.

Intersection of Normals

The point where two normals to a parabola intersect can satisfy special conditions, particularly when that intersection point lies on the parabola itself. Analyzing this intersection involves setting up the equations of the normals at different points and imposing conditions that allow the derivation of relationships among the parameters of the parabola and the distances between the points of contact where the normals were drawn.

Normal to a Parabola

For a parabola, the equation of the normal line can be derived by first finding the slope of the tangent and then using the negative reciprocal of this slope. Normals are significant in problems where intersections of normals are considered, as their behavior and intersection properties can lead to interesting geometric conclusions such as relationships among coordinates or distances between points on the parabola.

Tangent and Normal Lines

The tangent to a curve at a given point is the straight line that best approximates the curve near that point, and its slope is given by the derivative. The normal line at the same point is perpendicular to the tangent, meaning its slope is the negative reciprocal of the tangent’s slope. In the context of a parabola, determining the equations of these lines is fundamental for exploring reflective properties and geometric relationships.

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 standard equation, such as y² = 4ax, encapsulates its geometric properties including symmetry, the vertex, and the focal properties which are useful in various analyses involving tangents and normals.

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