If the normals at the points P, Q, R on the parabola y^2=4 ax intersect at the point (α, β), the

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If the normals at the points P, Q, R on the parabola y^2=4...

If the normals at the points $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ on the parabola $\mathrm{y}^{2}=4 \mathrm{ax}$ intersect at the point $(\alpha, \beta)$, then the orthocentre of triangle $\mathrm{PQR}$ is (a) $\left(\frac{2}{3} \mathrm{a}, 0\right)$ (b) $\left(\alpha-3 \mathrm{a}, \frac{-\beta}{2}\right)$ (c) $\left(\alpha-6 \mathrm{a}, \frac{-\beta}{2}\right)$ (d) $\left(\frac{\alpha}{2}, \frac{\beta}{2}\right)$

If the normals at the points $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ on the parabola $\mathrm{y}^{2}=4 \mathrm{ax}$ intersect at the point $(\alpha, \beta)$, then the orthocentre of triangle $\mathrm{PQR}$ is
(a) $\left(\frac{2}{3} \mathrm{a}, 0\right)$
(b) $\left(\alpha-3 \mathrm{a}, \frac{-\beta}{2}\right)$
(c) $\left(\alpha-6 \mathrm{a}, \frac{-\beta}{2}\right)$
(d) $\left(\frac{\alpha}{2}, \frac{\beta}{2}\right)$

Key Concepts

Parabola

A parabola is a conic section defined as the set of points equidistant from a fixed point, the focus, and a fixed line, the directrix. Its standard form, such as y² = 4ax, provides a framework to study its unique geometric properties, including key features like its vertex, axis of symmetry, and focus, which are essential when analyzing tangents and normals.

Normal to a Curve

A normal line to a curve at a given point is the line perpendicular to the tangent at that point. In analytic geometry, determining the equation of a normal is crucial for solving problems related to reflections, intersections, and distances, especially when the curve is a conic like a parabola, where explicit formulas for the normal can be derived.

Orthocentre of a Triangle

The orthocentre is the point where all three altitudes of a triangle intersect. It is an important concept in triangle geometry because it encapsulates significant information about the triangle's internal structure. Finding the orthocentre involves understanding how perpendiculars are dropped from the vertices to the opposite sides, making coordinate methods and slope relationships central to its determination.

Concurrency in Geometry

Concurrency occurs when three or more lines intersect at a single point. This concept is fundamental in many geometric settings, including the analysis of normals drawn to a conic section. Recognizing and utilizing the concurrency of lines such as normals helps in exploring deeper properties of the curves and triangles formed by points of intersection.

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