Two tangent lines l_1 and l_2 are drawn from the point (2,0) to the parabola 2 y^2=-x. If the lines l_1 and l_2 are also tangent to the circle (x-5)^2+y^2=r, then 17r is equal to ___________.

Given : y^2 = - x / 2 eqalign T equiv y = mx - 1 / 8m cr downarrow (2,0) cr => m^2 = 1 / 16 => m = ± 1 / 4 Tangents are y = 1 / 4x - 1 / 2, y = - x / 4 + 1 / 2 4y = x - 2 and 4y + x = 2 If these are also tangent to circle then d_c = r => | 5 - 2 / sqrt 17 | = sqrt r => r = ( 3 / sqrt 17 )^2 => 17r = 17 . 9 / 17 = 9

Two tangent lines l1 and l2 are drawn from the point 20... | null (Mathematics)

Given :

{y^2} = {{ - x} \over 2}

\eqalign{ & T \equiv y = mx - {1 \over {8m}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \downarrow (2,0) \cr}

\Rightarrow {m^2} = {1 \over {16}} \Rightarrow m = \, \pm \,{1 \over 4}

Tangents are

y = {1 \over 4}x - {1 \over 2},\,y = {{ - x} \over 4} + {1 \over 2}

4y = x - 2

and

4y + x = 2

If these are also tangent to circle then

{d_c} = r

\Rightarrow \left| {{{5 - 2} \over {\sqrt {17} }}} \right| = \sqrt r \Rightarrow r = {\left( {{3 \over {\sqrt {17} }}} \right)^2}

\Rightarrow 17r = 17\,.\,{9 \over {17}} = 9

Explanation

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