Two tangent lines $l_{1}$ and $l_{2}$ are drawn from the point $(2,0)$ to the parabola $2 \mathrm{y}^{2}=-x$. 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 ___________.
Answer (integer)
9
Solution
<p>Given : ${y^2} = {{ - x} \over 2}$</p>
<p>$$\eqalign{
& T \equiv y = mx - {1 \over {8m}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \downarrow (2,0) \cr} $$</p>
<p>$\Rightarrow {m^2} = {1 \over {16}} \Rightarrow m = \, \pm \,{1 \over 4}$</p>
<p>Tangents are $y = {1 \over 4}x - {1 \over 2},\,y = {{ - x} \over 4} + {1 \over 2}$</p>
<p>$4y = x - 2$ and $4y + x = 2$</p>
<p>If these are also tangent to circle then ${d_c} = r$</p>
<p>$$ \Rightarrow \left| {{{5 - 2} \over {\sqrt {17} }}} \right| = \sqrt r \Rightarrow r = {\left( {{3 \over {\sqrt {17} }}} \right)^2}$$</p>
<p>$\Rightarrow 17r = 17\,.\,{9 \over {17}} = 9$</p>
About this question
Subject: Mathematics · Chapter: Conic Sections · Topic: Parabola
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