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

$$\begin{gathered} \Rightarrow 4\sqrt{10}\sqrt{\left(2m+1\right)\left(m+2\right)}<0 \\ \Rightarrow m\in \left(-2,\frac{-1}{2}\right) \end{gathered}$$