Question

If the line \( y = mx \) does not intersect the circle \( (x + 10)^2 + (y + 10)^2 = 180 \), then a possible value of \( m \) is

• A. -3
• B. -4
• C. 1
• D. -1

Hint:

To ensure that a line does not intersect a circle, the perpendicular distance from the center of the circle to the line must exceed the radius.

Answer 1

The Correct Option is A

Step 1: Equation of the line and the circle.
The equation of the circle is \( (x + 10)^2 + (y + 10)^2 = 180 \), with center \( (-10, -10) \) and radius \( \sqrt{180} \). The equation of the line is \( y = mx \).
Step 2: Condition for no intersection.
For the line \( y = mx \) to not intersect the circle, the perpendicular distance from the center of the circle \( (-10, -10) \) to the line \( y = mx \) must be greater than the radius. The formula for the perpendicular distance from a point \( (x_1, y_1) \) to the line \( Ax + By + C = 0 \) is: \[ \text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}. \] Substitute the equation of the line \( y = mx \) as \( mx - y = 0 \).
Step 3: Calculate the distance.
The perpendicular distance from \( (-10, -10) \) to the line is: \[ \frac{|m(-10) - (-10) + 0|}{\sqrt{m^2 + (-1)^2}} = \frac{| -10m + 10 |}{\sqrt{m^2 + 1}}. \] This distance must be greater than \( \sqrt{180} \), i.e., the radius of the circle.
Step 4: Solve for \( m \).
After solving, we get that \( m = -3 \). Thus, the possible value of \( m \) is -3.