Q81 / 200 · TS EAMCET 2002 · Mathematics

The line $y=m x+c$ intercepts the circle $x^2+y^2=r^2$ in two distinct points, if

  • Correct answer
    A
    $-r \sqrt{1+m^2} < c < r \sqrt{1+m^2}$
  • B
    $c < -r \sqrt{1+m^2}$
  • C
    $c < r \sqrt{1+m^2}$
  • D
    None of the above

Solution

Equation of the circle is $$ x^2+y^2=r^2 $$ and the line is $m x-y+c=0$ The line (ii) intersect (i) in two distinct points, if the length of the $\perp^r$ from the centre $(0,0)$ to the line (ii) is less then $r$ Thus, $\left|\frac{0-0+c}{\sqrt{m^2+1}}\right| < r$ $$ \begin{array}{ccc} \Rightarrow & |c| < r \sqrt{m^2+1} \\ \Rightarrow & -r \sqrt{m^2+1} < c < r \sqrt{m^2+1} \end{array} $$

Hence, the correct answer is option: 'A'.

Relevant Questions