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  • Stuck here, help me understand: - If the parabolas and have a common normal, then which one of the following is a valid choice for t - Co-ordinate geometry - JEE Main

If the parabolas y^2 = 4b(x-c) and y^2 =8ax have a common normal, then which one of the following is a valid choice for the ordered triad (a,b,c)?

  • Option 1)

    (\frac{1}{2},2,3)

  • Option 2)

     

    (1,1,3)

  • Option 3)

     

    (1/2,2,0)

  • Option 4)

     

    (1,1,0)

Answers (1)

best_answer

 

Standard equation of parabola -

y^{2}=4ax

- wherein

 

 

Equation of tangent -

y= mx+\frac{a}{m}

- wherein

Tengent to y^{2}=4ax is slope form.

 

From the concept we have learnt ,

Normal to these two curves are

y = m(x -c ) -2bm -bm3

y = mx - 4am - 2am3

If they have common normal .

(c + 2b) m + bm3 = 4am + 2am3

= (4a - c -2b)m = (b-2a)m3

=(4a - c -2b) = (b-2a)m2

m^{2} = \frac{c}{2a-b} -2 > 0

\frac{c}{2a-b} > 0


Option 1)

(\frac{1}{2},2,3)

Option 2)

 

(1,1,3)

Option 3)

 

(1/2,2,0)

Option 4)

 

(1,1,0)

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