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Parabola question

2024 · 27 Jan · Shift 1 · Q34
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  5. /2024 · 27 Jan · Shift 1 · Q34

Parabola question

2024 · 27 Jan · Shift 1 · Q34

JEE MainMathematicsParabolaMCQ+4 / −1
If the shortest distance of the parabola y2=4xy^2=4 xy2=4x from the centre of the circle x2+y2−4x−16y+64=0x^2+y^2-4 x-16 y+64=0x2+y2−4x−16y+64=0 is d\mathrm{d}d, then d2\mathrm{d}^2d2 is equal to :
  1. A
    16
  2. B
    24
  3. C
    20
  4. D
    36
View written solutionFree

Correct answer: C

  1. Find the centre of the circle

Given circle: x2+y2−4x−16y+64=0x^2+y^2-4x-16y+64=0x2+y2−4x−16y+64=0

Complete squares: x2−4x+y2−16y+64=0x^2-4x+y^2-16y+64=0x2−4x+y2−16y+64=0 x2−4x+4+y2−16y+64−4=0x^2-4x+4+y^2-16y+64-4=0x2−4x+4+y2−16y+64−4=0 (x−2)2+(y−8)2−4=0(x-2)^2+(y-8)^2-4=0(x−2)2+(y−8)2−4=0 (x−2)2+(y−8)2=4(x-2)^2+(y-8)^2=4(x−2)2+(y−8)2=4

So the centre of the circle is C=(2,8).C=(2,8).C=(2,8).

We need the shortest distance from C=(2,8)C=(2,8)C=(2,8) to the parabola y2=4x.y^2=4x.y2=4x.

  1. Take a general point on the parabola

For the parabola y2=4xy^2=4xy2=4x, a standard parametric point is P=(t2,2t).P=(t^2,2t).P=(t2,2t).

Distance squared from C=(2,8)C=(2,8)C=(2,8) to PPP is D2=(t2−2)2+(2t−8)2.D^2=(t^2-2)^2+(2t-8)^2.D2=(t2−2)2+(2t−8)2.

Now expand: D2=t4−4t2+4+4t2−32t+64D^2=t^4-4t^2+4+4t^2-32t+64D2=t4−4t2+4+4t2−32t+64 D2=t4−32t+68.D^2=t^4-32t+68.D2=t4−32t+68.

We must minimize f(t)=t4−32t+68.f(t)=t^4-32t+68.f(t)=t4−32t+68.

  1. Minimize the distance squared

Differentiate: f′(t)=4t3−32=4(t3−8).f'(t)=4t^3-32=4(t^3-8).f′(t)=4t3−32=4(t3−8).

Set equal to zero: 4(t3−8)=04(t^3-8)=04(t3−8)=0 t3=8t^3=8t3=8 t=2.t=2.t=2.

Second derivative: f′′(t)=12t2>0f''(t)=12t^2>0f′′(t)=12t2>0 for t=2t=2t=2, so this gives a minimum.

  1. Compute the minimum value

d2=f(2)=24−32(2)+68=16−64+68=20.d^2=f(2)=2^4-32(2)+68=16-64+68=20.d2=f(2)=24−32(2)+68=16−64+68=20.

Hence, d2=20.\boxed{d^2=20}.d2=20​.

  1. Check with options

Options are:

  • A: 161616
  • B: 242424
  • C: 202020
  • D: 363636

So the correct option is C.\boxed{\text{C}}.C​.

  1. Comparison with stored correct answer

Stored correct answer: C

Our derived answer: C

They match.

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