Help with question please (1 Viewer)

[dabatman]

Hi,

Can anyone help me with this question? (With part a only)

 

[Drongoski]

For:

a) circle's centre is (0,0) and radius sqrt(5)

distance of centre from given line is (0-0+16)/sqrt(34) = 2.74 ... > sqrt(5)
.: line cuts circle at zero places

 

[andrew12678]

Use perpendicular distance formula with the Line given and the centre of the circle, if it is larger than the radius of the circle then it will intersect no times, if it equal to the radius of circle it will be a tangent and intersect once, if it is less than the radius then it will intersect twice

 

[Drongoski]

For:

a) circle's centre is (0,0) and radius sqrt(5)

distance of centre from given line is (0-0+16)/sqrt(34) = 2.74 ... > sqrt(5)
.: line cuts circle at zero places

b) circle's centre is (1,5) and radius sqrt(10)

distance of centre from given line = |3-5-8|/sqrt(10) = sqrt(10)
.: line just touches circle and so intersects circle at 1 point

 

[dabatman]

Can anyone help me with part c please

 

[Drongoski]

Can anyone help me with part c please

For c):

You solve simultaneously for the eqns of the straight line y = mx and of the circle. You get a quadratic eqn in x as follows:



Since the line touches the circle, this eqn has a repeated root (if line cuts thru the circle, you have 2 different roots) and therefore the discriminant of this quadratic eqn = "b² - 4ac" = 0;
i.e. (-2m-6)^2 - 4 x (1+m^2) x 6 = 0
i.e.4(m^2 +6m + 9) - 4 x 6 x (1++m^2) = 0; i.e 5m^2 - 6m -3 = 0