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Notes On Equation Of Chord Of A Parabola

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Published in: Mathematics
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Equation of Chord of a Parabola When the midpoint is Given

Abdusamad / Dubai

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Qualification: B.Tech Mechanical Engineering

Teaches: Mathematics, Maths

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  1. Equation of Chord of a Parabola When Midpoint is given Chord of a parabola is a line segment whose end points lie on the parabola. The chord of a parabola which passes through the focus is called focal chord and the chord passing through the focus and perpendicular to the axis of the parabola is called latus rectum. Chord PQ R (xt,y,) Q(xvY3) Figure 1: chord of parabola y = 4m; Consider a parabola y and a chord PQ whose midpoint is R (x„yl), as shown in the figure Let P (X2, (X; , be the coordinates of P and Q Then using the midpoint formula Xl and x2 can be written as follows 2 + X3 = 2Xl ......(2) 2 Also since P (x2, Y2) , Q (.13, yo are the points on the parabola Y' We can write
  2. Y2 — Subtracting (3) from (4) 2 Y2 = 4a(X3 (Since h + Y2 Where (Y3 —Y2) = 2)'1 from (2)) is the slope of the chord PQ , which is mpQ equal to The equation of chord PQ can be written as: y — = mpQ ) _yyl — y 1 — 2ar— 2aXl 2 axl (Subtracting on both sides) y12 — 4aVl So, the equation of chord of the parabola y =4axwith midpoint yyl —2a(x+ xo = y] 2 —4aXl
  3. Illustration Find the equation of chord of the parabola y chord Solution: = 8X such that the focus of the parabola bisect the It is given that the chord bisect at the focus of the parabola Therefor the midpoint of the chord is (a, O) By comparing Y' and the standard form y = 4ax Hence the midpoint of chord is (2, O) The equation of chord of parabola with given midpoint is given by yyl —2a(x+ xo = —4aXl Substitute Xl = 2, y, = 2 in the above equation We get yxO-2x2(X+2) =o-4x2x2 — —16 Hence x=2 is the required equation of chord Illustration Find the locus of the midpoint of the chord passes through the vertex of the parabola Y' Solution:
  4. The parametric form of y = 4av is x=2at ,y=2at Let the point (h,k) be the midpoint of the chord passing through the vertex of the parabola Y2 4ar so, 2 at 2 ......(2) 2 From (l) and (2) 2 e = 2ah Replace the point (h,k) by (x, y) y = 2ax which is the required locus the midpoint of the all chords passing through the vertex of given parabola
  5. N/A

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