# Given : A circle,   and a parabola, Statement - I : An equation of a common tangent to these curves is Statement - II : If the line , is their common tangent, then m satisfies . Option 1) Statement - I is false ; Statement - II is true.   Option 2) Statement - I is ture ; Statement - II is true ; Statement - II is a correct explanation for statement - I.     Option 3) Statement - I is ture ; Statement - II is true ; Statement - II is not a correct explanation for statement - I.     Option 4) Statement - I is ture ; Statement - II is false.

As we learnt in

Condition of tangency -

$c^{2}=a^{2}\; (1+m^{2})$

- wherein

If  $y=mx+c$  is a tangent to the circle $x^{2}+y^{2}=a^{2}$

and

Equation of tangent -

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

- wherein

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

$x^{2}+y^{2}=\frac{5}{2}\:and\:y^{2}=4\sqrt{5x}$

Tangent to circle is $y=mx+\frac{5}{2}\sqrt{1+m^{2}}$

Tangent to parabola is $y=mx+\frac{\sqrt{5}}{m}$

So, $\frac{5}{2}\sqrt{1+m^{2}}=\frac{\sqrt{5}}{m}$

On solving m=1

Thus tangent is $y=x+\sqrt{5}$

Option 1)

Statement - I is false ; Statement - II is true.

This option is incorrect.

Option 2)

Statement - I is ture ; Statement - II is true ; Statement - II is a correct explanation for statement - I.

This option is incorrect.

Option 3)

Statement - I is ture ; Statement - II is true ; Statement - II is not a correct explanation for statement - I.

This option is correct.

Option 4)

Statement - I is ture ; Statement - II is false.

This option is incorrect.

N

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