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Parabolic Curve

The equation y = Ax2 + Bx + C always represents parabola and is generally called parabolic curve. Its vertex is at (h, k) = 70915.png axis parallel to the y-axis and latus rectum = 70909.png and the curve opening upward and downward according to A > 0 or A < 0 respectively.
Position of a point with respect to a parabola y2 = 4ax Now P(h, k) will lie outside, on or inside the parabola y2 = 4ax accordingly as (k2 – 4ah) >, =, < 0.

Focal chord and its properties

Any point on the parabola y2 = 4ax can be taken as (at2, 2at), where t is a parameter and t R. Any line passing through the focus of the parabola is called focal chord of the parabola.
  1. If chord joining P ≡ (at12, 2at1) and Q ≡ (at22, 2at2) is focal chord then t1t2 = –1.
  2. If point P is (at2, 2at), then length of focal chord PQ is a(t + 1/t)2.
  3. The length of the focal chord which makes an angle θ with x positive direction of the x-axis is 4a cosec2 θ.
  4. Semi-latus rectum is harmonic mean of SP and SQ, where P and Q are extremities of latus rectum.
  5. Circle described on any focal distance as a diameter always touches tangent at vertex.
  6. Circle described on any focal chord as a diameter always touches the directrix.

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