LOCUS & PARABOLAS: See questions below?

P(16,16) and Q (-4,1) are points on the parabola x^2=16y

(i) Show that PQ is a focal chord

(ii) Find the equations of the tangents at P and Q

(iii) Show that these 2 tangents are perpendicular

(iv) Show that the 2 tangents intersect on the directrix

Full working put, please. And thanks for helping out!

2 Answers

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  • 2 years ago
    Favorite Answer

    (i)

    First find the focal point of the parabola y = x²/16

    Focus: (h,k+1/4a)

    In vertex form y = a(x−h)²+k, the parabola in question is expressed as y = ¹/₁₆(x−0)²+0,

    so the vertex (h,k) is at (0,0) and a=¹/₁₆

    Focus: (0,0+1/4a) = (0,4)

    Let F be the focus. If PQ is a focal chord, then PQ has the same slope as PF (or QF)

    Slope PQ: ∆y/∆x = ¾

    Slope PF: ∆y/∆x = ¾

    Slope QF: ∆y/∆x = ¾

    -----------> PQ and PF (and QF) have the same slope which shows that PQ is a focal chord

    (ii)

    dy/dx = x/8

    The derivative at the point is the slope at that point:

    Slope at P: 16/8 = 2

    Slope at Q: −½

    (y−y₀) = m(x−x₀)

    y = mx−mx₀+y₀

    ----------->

    Tangent line at P: y = 2x−2*16+16 = 2x−16

    Tangent line at Q: y = −½x−(−½)*(−4)+1 = −½x−1

    (iii)

    Two perpendicular lines satisfy the condition: m₁m₂ = −1

    Slope P = 2

    Slope Q = −½

    2 * −½ = −1

    -----------> Tangent lines of P and Q are perpendicular since the product of their slopes equals −1

    (iv)

    The vertex is the midpoint of the line joining the focus and the intersection of the directrix with

    the axis of symmetry.

    Vertex: (0,0)

    Focus: (0,4)

    Axis of symmetry: x=0

    Intersection of directrix and axis of symmetry is then (0,−4), if (0,0) is the midpoint

    => Directrix: y = −4

    The tangent lines of P and Q intersect on the directrix IFF they both have the same

    x-coordinate at y = −4

    Tangent at P: −4 = 2x−16 => x = 6

    Tangent at Q: −4 = −½x−1 => x = 6

    -----------> Tangent lines of P and Q intersect on the directrix at the point (6,−4)

  • DWRead
    Lv 7
    2 years ago

    That doesn't make sense. The endpoints of the focal chord are (-8,4) and (8,4)

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