(b) Focal chord : A chord of the parabola, which passes through the focus is called a focal chord. Let the parabola we consider and draw chords be y2 = 4ax. Hence locus of (α, β) is y2 = 2a(x – a). Midpoint of OP is (at2/2,at). (ii) focus is (a,0) (iii) Axis is y = 0 (iv) Directrix is x + a = 0 (a) Focal distance : The distance of a point on the parabola from the focus is called the focal distance of the point. y + tx = 2at + at 3. Find the locus of middle points of a family of focal chords of the parabola y^2=4ax Class: 11 2 See answers rohitkumargupta rohitkumargupta HELLO DEAR , Let the equation of the parabola be y2 = 4ax. The locus of the middle points of all chords of the parabola y 2 = 4ax passing through the vertex is. Illustration : Through the vertex O of a parabola y 2 = 4x chords OP and OQ are drawn at right angles to one another. a parabola. QUESTION: 13. The locus of the mid point of the focal radii of a variable point moving on the parabola, y 2 =4ax is a parabola whose (A) vertex is (a/2,0) (B) Latus rectum is half the latus rectum of the original parabola (C) Focus has the co-ordinates (a,0) (D) Directrix is y-axis 2. A circle drawn on any focal chord of the parabola y2=4ax as diameter cuts parabola at two points ‘t’ and ‘t ’ (other than the extrimity of focal chord) the (a) tt = –1 (b) tt = 2 1. … (1) Equation of the chord with mid point (x 1, y 1) is T = S1. Also find the locus of the middle point of PQ. Then the midpoint of the chord PQ. Show that as the focal chord varies, the line PQ passes through a ﬁxed point. Show that for all position of P, PQ cuts the axis of the parabola at a fixed point. Solution: Equation of the normal chord at any point (at 2, 2at) of the parabola is. or yy 1 – 2a(x + x 1) = y 1 2 – 4ax 1 or yy 1 – 2ax = y 1 2 – 2ax 1. Let M(p,q) be the midpoint of the chord OP. The locus of the mid-points of the focal chord of the parabola y 2 = 4ax is. For this parabola : (i) Vertex is (0,0). The Vertex is O(0.0), which is one end of the chord. ... We have y 2 = 4 ax We know that ends of focal chord are at 2, 2 at and a t 2,-2 a t Let h, k be the mid point ... 2 h = at 2 + a t 2 ⇒ 2 h a = t 2 + 1 t 2 ⇒ 2 h a = t-1 t 2 + 2 a 2 + b … (Chords orthogonal at the vertex) Let PQ be a chord of a parabola with vertex O such that angle POQ is a right angle. Locus of midpoint of focal chord. a straight line. This worksheet shows the locus of the midpoint of the focal chord of a parabola The extremities of a focal chord of the parabola y 2 = 4ax may be taken as the points t and − 1/t. If the parabola is Y 2 = 4ax take the focal chord which is easy for calculation e.x. Let the other end be a varaible point P given by (at2,2at). Find the locus of the middle point of focal chord of parabola whose eq. 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