A circle of radius a, centre (0,0) touches the directrix of...

A circle of radius a, centre $(0,0)$ touches the directrix of $y^{2}=4 a x$ at $P$. Tangents are drawn from $P$ to the parabola touches it at $\mathrm{Q}$ and $\mathrm{R}$. Then (a) othocenter of $\triangle \mathrm{PQR}$ is $(\mathrm{a}, 0)$ (b) othocenter of $\Delta \mathrm{PQR}$ is $(-\mathrm{a}, 0)$ (c) circumcenter of $\Delta \mathrm{PQR}$ is $(\mathrm{a}, 0)$ (d) circumcenter of $\Delta \mathrm{PQR}$ is $(-\mathrm{a}, 0)$

A circle of radius a, centre $(0,0)$ touches the directrix of $y^{2}=4 a x$ at $P$. Tangents are drawn from $P$ to the parabola touches it at $\mathrm{Q}$ and $\mathrm{R}$. Then
(a) othocenter of $\triangle \mathrm{PQR}$ is $(\mathrm{a}, 0)$
(b) othocenter of $\Delta \mathrm{PQR}$ is $(-\mathrm{a}, 0)$
(c) circumcenter of $\Delta \mathrm{PQR}$ is $(\mathrm{a}, 0)$
(d) circumcenter of $\Delta \mathrm{PQR}$ is $(-\mathrm{a}, 0)$

Key Concepts

Orthocenter

The orthocenter of a triangle is the common intersection point of the three altitudes of the triangle. It is one of the notable centers of a triangle, revealing essential geometric properties and relationships among the triangle’s sides and angles.

Circumcenter

The circumcenter is the point that is equidistant from all three vertices of a triangle. It serves as the center of the triangle's circumscribed circle (circumcircle) and is found by intersecting the perpendicular bisectors of the triangle's sides.

Tangent Lines

Tangent lines to a curve are lines that intersect the curve at precisely one point and share the same instantaneous direction (slope) at the point of contact. In the study of conic sections and circles, tangents are used to explore geometric relationships, such as distances and angles, and to locate key points on the figures.

Circle Tangency

Circle tangency refers to the condition where a circle touches a line (or another curve) at exactly one point. The property that a radius drawn to the point of tangency is perpendicular to the tangent line is fundamental in solving problems involving intersections between circles and lines.

Directrix

The directrix is a fixed line used in the definition of a conic section like a parabola. For a parabola, every point on the curve has equal distance to the focus and this line, playing a crucial role in determining the curve's geometric properties and related tangency conditions.

Parabola

A parabola is a conic section defined as the set of all points equidistant from a fixed point, called the focus, and a fixed line, called the directrix. Its geometry includes distinct properties such as reflective behavior, the standard form of equations, and the nature of its tangents.

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