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Engineering
1 week, 5 days ago

Can someone help me with this, - Co-ordinate geometry - JEE Main-6

Statement 1: An equation of a common tangent to the parabolay^{2}= 16\sqrt{3}x  and the ellipse 2x^{2}+y^{2}= 4 is y= 2x+2\sqrt{3}

Statement 2: If the line y= mx+\frac{4\sqrt{3}}{m},(m\neq 0) is a common tangent to the parabola

y^{2}= 16\sqrt{3}x and the ellipse 2x^{2}+y^{2}= 4, then m satisfies  m^{4}+2m^{2}= 24

  • Option 1)

    Statement 1 is false, statement 2 is true

  • Option 2)

    Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1

  • Option 3)

    Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

  • Option 4)

    Statement 1 is true, statement 2 is false

 
Answers (1)
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S solutionqc
Answered 1 week, 5 days ago

As we learnt in

Equation of tangent -

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

- wherein

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

 

 

 

Statement 1 :

y^{2}= 16\sqrt{3}x,y= mx+\frac{4\sqrt{3}}{m}

\frac{x^{2}}{2}+\frac{y^{2}}{4}= 1,x= m_{1}y+\sqrt{4m^{2}_{1}+2}

\Rightarrow y= \frac{x}{m_{1}}-\sqrt{4+\frac{2}{{m_{1}}^{2}}},m= \frac{1}{m_{1}}

Now\: \: \left ( \frac{4\sqrt{3}}{m} \right )^{2}= \left ( -\sqrt{4+\frac{2}{{m_{1}}^{2}}} \right )^{2}

 

\Rightarrow \frac{48}{m^{2}}= 4+\frac{2}{{m_{1}}^{2}}= 4+2m^{2}\Rightarrow \frac{24}{m^{2}}= 2+m^{2}

\Rightarrow m^{4}+2m^{2}-24=0\cdots \cdots \cdots \cdots (i)

\Rightarrow \left ( m^{2}+6 \right )\left ( m^{2}-4 \right )= 0\Rightarrow m\pm 2

Statement 2 :

If y=mx +\frac{4\sqrt{3}}{m} is a common tangent to y^{2}= 16\sqrt{3}x and ellipse 2x^{2}+y^{2}= 4\: \: then \: \: m\: \: satisfies m^{4}+2m^{2}-24=0

From (i)     statement 2 is a correct explanation for statement 1.

 

 


Option 1)

Statement 1 is false, statement 2 is true

This option is incorrect.

Option 2)

Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1

This option is correct.

Option 3)

Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

This option is incorrect.

Option 4)

Statement 1 is true, statement 2 is false

This option is incorrect.

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