Question

If the line y = mx does not intersect the circle (x + 10)² + (y + 10)² = 180, then write the set of values taken by m.

Answer 1

Let us put y = mx in the equation (x + 10)² + (y + 10)² = 180.
Now, we have:
(x + 10)² + (mx + 10)² = 180
On simplifying, we get: 

\[x^2 \left( m^2 + 1 \right) + 20x\left( m + 1 \right) + 20 = 0\]

∴ Discriminant (D) = \[\sqrt{400 \left( m + 1 \right)^2 - 80( m^2 + 1)} = 4\sqrt{10}\sqrt{\left( 2m + 1 \right)\left( m + 2 \right)}\] It is given that the line y = mx does not intersect the circle (x + 10)² + (y + 10)² = 180.
∴ D <  0 

\[\Rightarrow 4\sqrt{10}\sqrt{\left( 2m + 1 \right)\left( m + 2 \right)} < 0\]

\[ \Rightarrow m \in \left( - 2, \frac{- 1}{2} \right)\]