If a normal chord of y^2=4 x subtends a right angle at the vertex, then the extremities of the c

step 1 So in this question it is given that a normal chord of this parabola y square equals 4x subtend

If a normal chord of y^2=4 x subtends a right angle at the...

If a normal chord of $\mathrm{y}^{2}=4 \mathrm{x}$ subtends a right angle at the vertex, then the extremities of the chord are (a) $(3, \pm 2 \sqrt{2}),(2, \pm 4 \sqrt{2})$ (b) $(2, \pm 2 \sqrt{2}),(8, \mp 4 \sqrt{2})$ (c) $(3,4 \sqrt{2}),(2, \sqrt{2})$ (d) $(3, \pm 2 \sqrt{2}),(8, \mp 4 \sqrt{2})$

If a normal chord of $\mathrm{y}^{2}=4 \mathrm{x}$ subtends a right angle at the vertex, then the extremities of the chord are
(a) $(3, \pm 2 \sqrt{2}),(2, \pm 4 \sqrt{2})$
(b) $(2, \pm 2 \sqrt{2}),(8, \mp 4 \sqrt{2})$
(c) $(3,4 \sqrt{2}),(2, \sqrt{2})$
(d) $(3, \pm 2 \sqrt{2}),(8, \mp 4 \sqrt{2})$

Key Concepts

Subtended Angle at a Point

When a chord of a curve subtends an angle at a particular point (for instance, the vertex of the parabola), it means that the two lines drawn from the point to the endpoints of the chord form that given angle. Analyzing such subtended angles involves using vector or slope methods in analytic geometry and is key in setting up equations to find the coordinates of the endpoints.

Parabola

A parabola is a type of conic section defined as the locus of all points that are equidistant from a fixed point (the focus) and a fixed line (the directrix). Its standard equation is useful in analytical geometry, and understanding its properties such as the vertex, axis, and focus can be essential in solving geometric problems involving parabolic curves.

Chord

A chord of a curve is a line segment with both endpoints lying on the curve. In the context of a parabola, a chord may have special properties such as bisecting angles or being parallel to the axis of the parabola, and analyzing these chords can lead to conditions that determine specific positions or relationships among points on the curve.

Normal to a Parabola

A normal to a parabola is a line that is perpendicular to the tangent at a given point on the curve. The normal line’s equation can be derived using differential calculus, and its properties are instrumental in many problems related to reflections, orthogonality, and finding intersections with other curves or with the parabola itself.

Right Angle Condition

The condition where a chord subtends a right angle (90 degrees) at a given point introduces an orthogonality criterion. This condition implies that the product of the slopes of the lines joining the point to the endpoints is -1. It is a powerful tool in coordinate geometry for establishing relationships between the coordinates of the endpoints and for solving for unknowns in geometric problems.

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