Question

- m - 1 = 0" align="absmiddle" class="latex-formula">
and
${x}^{2} + {y}^{2} - 4x - 2y + 1 = 0$
do not intersect.

Find possible range values for m.​

Answer 1

Hey there mate :)

As per question, two equations are given.

We have to find the range of possible values of m so that both equations do not intersect.

If they intersect, both equations have to be satisfied by the same pair of (x,y) values.

We can then write :-

$mx - y - m - 1 = 0 \\ = > m(x - 1) - y - 1 = 0 \\ = > y + 1 = m(x - 1)$

This is an equation of a line that passes through point (1,-1) and has a slope of m.

We can then rearrange the other equation as :-

${x}^{2} + {y}^{2} - 4x - 2y + 1 = 0$

$= > {x}^{2} - 2 \times 2x + 4 - 4 + y^{2} - 2 \times y + 1 = 0$

$= > (x - 2)^{2} + (y - 1)^{2} - 4 = 0$

$= > (x - 2)^{2} + (y - 1)^{2}$

This is the equation of a circle with radius $\sqrt{4} = 2$and center at (2,1)

Then, we have m values where :-

1. Line intersects circle at two points

2. Line is tangent to circle at one point.

3. Line does not intersect the circle.

The interval of m where m does not intersect the circle will be between the two values of m where the line is a tangent to the circle, which happens at two points with different values of m.