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Evaluate, min[(x1-x2)^2+(5+sqrt(1-x1^2)-...

Evaluate, `min[(x_1-x_2)^2+(5+sqrt(1-x_1^2)-sqrt(4x_2))^2],AAx_1,x_2 in R ,` is

A

`4sqrt(1)`

B

`3-2sqrt(2)`

C

`sqrt(5)+1`

D

`4sqrt(5)-1`

Text Solution

Verified by Experts

The correct Answer is:
B

(2) Let `y_(1)=3+sqrt(1-x_(1)^(2))andy_(2)=sqrt(4x_(2))`
`orx_(1)^(2)+(y_(1)-3)^(2)=1andy_(2)^(2)=4x_(2)`
Thus, `(x_(1),y_(1))` lies on the circle `x^(2)+(y-3)^(2)=1`.
Also, `(x_(2),y_(2))` lies on the upper half of the parabola `y^(2)=4x`.
Thus, the given expression is square of the shortest distance between the curves `x^(2)+(y-3)^(2)=1andy^(2)=4x`.
Now, the shortest distance always occurs along the common normal to the curves and the normal to the circle passes through the center of the circle.
Normal to the parabola `y^(2)=4x" is "y-mx-2m-m^(3)`. It passes through (0,3).
Therefore, `m^(3)+2m+3=0`, which has only one root, m=-1.
Hence, the required minimum distance is `sqrt(1+1)-1=sqrt(2)-1`.
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CENGAGE PUBLICATION-PARABOLA-EXERCISE (SINGLE CORRECT ANSWER TYPE )
  1. The equation of the common tangent to the parabolas y^2= 4ax and x^2=...

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  2. If line P Q , where equation is y=2x+k , is a normal to the parabola w...

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  3. Evaluate, min[(x1-x2)^2+(5+sqrt(1-x1^2)-sqrt(4x2))^2],AAx1,x2 in R , ...

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  4. If the normals to the parabola y^2=4a x at three points (a p^2,2a p), ...

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  5. Normals A O ,A A1a n dA A2 are drawn to the parabola y^2=8x from the p...

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  6. If the normals to the parabola y^2=4a x at the ends of the latus rectu...

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  7. From a point (sintheta,costheta), if three normals can be drawn to the...

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  8. If the normals at P(t(1))andQ(t(2)) on the parabola meet on the same p...

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  9. If the normals to the parabola y^2=4a x at P meets the curve again at ...

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  10. PQ is a normal chord of the parabola y^2 =4ax at P, A being t...

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  11. P ,Q , and R are the feet of the normals drawn to a parabola (y-3)^2=8...

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  12. Normals at two points (x1y1)a n d(x2, y2) of the parabola y^2=4x meet ...

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  13. The endpoints of two normal chords of a parabola are concyclic. Then ...

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  14. If normal at point P on the parabola y^2=4a x ,(a >0), meets it again ...

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  15. The set of points on the axis of the parabola (x-1)^(2)=8(y+2) from wh...

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  16. Tangent and normal are drawn at the point P-=(16 ,16) of the parabola ...

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  17. In parabola y^2=4x, From the point (15,12), three normals are drawn th...

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  18. The line x-y=1 intersects the parabola y^2=4x at A and B . Normals at ...

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  19. If normal are drawn from a point P(h , k) to the parabola y^2=4a x , t...

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  20. The circle x^(2)+y^(2)+2lamdax=0,lamdainR, touches the parabola y^(2)=...

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