# The circle $\dpi{100} x^{2}+y^{2}=4x+8y+5$  intersects the line $\dpi{100} 3x-4y=m$ at two distinct points if Option 1) $-85< m< -35$ Option 2) $-35< m< 15$ Option 3) $15< m< 65$ Option 4) $35< m< 85$

P Prateek Shrivastava

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 $x^{2}+y^{2}=a^{2}$

$\\ x^{2}+y^{2}-4x-8y-5=0 \: \\ \\ and \: 3x-4y=m \\ \\ x=\frac{m+4y}{3}$

$\frac{\left ( m+4y \right )^{2}}{9}+y^{2}-\frac{(4m+16y)}{3}-8y-5\Rightarrow 0$

$\frac{25}{9}y^{2}+y\left ( \frac{8m}{9}-\frac{40}{3} \right )+\left ( \frac{m^{2}}{9}-5 \right )\Rightarrow 0$

${25}y^{2}+8y\left ( m-120 \right )+\left ( m^{2}-45 \right )\Rightarrow 0$

$\\ D> 0 \\ \\ 8^{2}\left ( m-120 \right )^{2}-4\times 25\left ( m^{2}-45 \right )\Rightarrow 0$

$16m^{2}-32\times 120m+16\times 120^{2}-25m^{2}+25\times 45> 0$

$-9m^{2}-32\times 120m+16\times 120^{2}+25 \times45> 0$

On solving, we get

$\\ \left ( m+35 \right ) (m-15)< 0 \\ \\ m\epsilon \: (-35, 15)$

Option 1)

$-85< m< -35$

THis option is incorrect

Option 2)

$-35< m< 15$

THis option is correct

Option 3)

$15< m< 65$

THis option is incorrect

Option 4)

$35< m< 85$

THis option is incorrect

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