Focal chord length of parabola
WebNov 20, 2013 · This is the length of the focal chord (the "width" of a parabola at focal level). Let x 2 = 4 p y be a parabola. Then F ( 0, p) is the focus. Consider the line that passes through the focus and parallel to the directrix. Let A and A ′ be the intersections of the line and the parabola. Then A ( − 2 p, p), A ′ ( 2 p, p), and A A ′ = 4 p. Share Cite WebAnswer: Consider the parabola: The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The "latus rectum" is the chord of the parabola which is parallel to the directrix and passes through the focus. In fact the “latus rectum” used to be calle...
Focal chord length of parabola
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Web(v) Length of the focal chord having t 1 and t 2 as end points is a (t 1 — t 1) 2. (vi) Chord of contact drawn from a point (x 1, y 1) to the parabola y 2 = 4ax is yy 1, = 2a (x + x 1) (vii) Equation of the chord of the parabola y 2 = 4ax, which is bisected at (x 1 , y 1) is given by T = S 1 i.e. , yy 1 — 2a (x + x 1) = y 12 – 4ax WebDec 8, 2024 · Question 4 :$$ $$ Let PQ be a focal chord of a parabola with origin as a focus . Coordinates of point P and Q be (-2,0) and (4,0) respectively . Find length of latus rectum and equation of tangent at vertex of parabola.
WebThe length of a focal chord of the parabola y 2=4ax at a distance b from the vertex is c. Then A 2a 2=bc B a 3=b 2c C ac=b 2 D b 2c=4a 3 Hard Solution Verified by Toppr Correct option is D) Equation of the focal line passing through (a,0) is y=m(x−a) The distance of this line from the vertex is b. ⇒b= ∣∣∣∣∣ 1+m 2am ∣∣∣∣∣ ⇒b 2(1+m 2)=a 2m 2 .... (1)
WebMar 14, 2024 · Consider a parabola y 2 = 4 a x , parameterize it as x = a t 2 and y = 2 a t, then it is found that if we have a line segment passing through focus, with each points having value of t as t 1 and t 2 for the parameterization, then it must be that: t 1 ⋅ t 2 = − 1 Hope for hints. conic-sections Share Cite Follow edited Mar 14, 2024 at 15:05 WebSimplifying gives us the formula for a parabola: x 2 = 4py In more familiar form, with " y = " on the left, we can write this as: \displaystyle {y}=\frac { {x}^ {2}} { { {4} {p}}} y = 4px2 where p is the focal distance of the parabola. Now let's see what "the locus of points equidistant from a point to a line" means.
WebApr 11, 2024 · We are given a parabola \[{y^2} = 4ax\] Let us assume that the chord cuts the X-axis at point D(a,0) Then according to the question we are given the shortest distance from center to the chord is b. Length of the focal chord is c. The distance \[OD = a\]. Let us assume the focal chord makes an angle x with the X-axis.
WebThe length of a focal chord of the parabola y2 =4ax at a distance ‘b’ from the vertex is ‘c’, then A 2a2=bc B a3=b2c C b2 =ac D b2c=4a3 Solution The correct option is D b2c =4a3 Let the angle made by focal chord with x – axis be θ ∴ sinθ= b a Length of focal chord, c =4acosec2θ ⇒ c= 4a(a b)2 ⇒ b2c =4a3 Suggest Corrections 28 Similar questions Q. birgit whitmanWebThe length of this focal chord of an ellipse is the focal length of that ellipse. The formula to calculate the focal length of the ellipse whose equation is x² / a² + y² / b² = 1 with the condition that the ellipse is inclined to the major axis at … birgit witamwasWebParabola (TN) - Free download as PDF File (.pdf), Text File (.txt) or read online for free. 1ST LECTURE 1. CONIC SECTIONS : A conic section, or conic is the locus of a point which moves in a plane so that the ratio of its distance from a fixed point to its perpendicular distance from a fixed straight line is a constant i.e. PS = constant = e. dancing fry sensoryWebSolution The correct option is A (8, –8) For the parabola y2 = 8x; focus S (2, 0). Given point is P (1 2,2) Slope of ←→ SP is 2−0 1 2−2 = −4 3 Equation to ←→ SP is4x+3y−8= 0 4x+3y−8= 0⇒ 4x=8−3y Substituting this value of 4x in y2 = 8x we get y2 = 2(8−3y) ⇒y2+6y−16−16 =0 ⇒(y+8)(y−2) = 0 ⇒ y= 2or−8 y =−8 ⇒4x =8−3(−8)= 32⇒ x= 8 ∴ point … birgit wimmer novomaticWebThe length of a focal chord of the parabola y2 =4ax at a distance ‘b’ from the vertex is ‘c’, then A 2a2=bc B a3=b2c C b2 =ac D b2c=4a3 Solution The correct option is D b2c =4a3 … birgit weyhe madgermanesWebThe length of the intercept on the normal at the point (a t 2, 2 a t) of the parabola y 2 = 4 a x made by the circle which is described on the focal distance of the given point as diameter is. Hard. View solution > If the tangent and normals at the extremities of a focal chord of a parabola intersect at (x 1 ... birgit weldishoferWebThe latus rectum of a parabola is the chord that is passing through the focus of the parabola and is perpendicular to the axis of the parabola. The latus rectum of parabola can also be understood as the focal chord which is parallel to the directrix of parabola.The length of latus rectum for a standard equation of a parabola y 2 = 4ax is equal to LL' = 4a. dancing fruits and veggies