1. length of AB

the line $y = mx$ is a tangent to the circle $x^2 + y^2 - 10y + 16 = 0$
a) find the two possible values of m
b) the tangents meet the circle at points A and B. find the length of AB

i've done a) and its definitely $m = -\frac {4}{3}$ and $\frac {4}{3}$

but i don't know how to tackle b), could someone show me please? thankyou

2. Originally Posted by mark
the line $y = mx$ is a tangent to the circle $x^2 + y^2 - 10y + 16 = 0$
a) find the two possible values of m
b) the tangents meet the circle at points A and B. find the length of AB

i've done a) and its definitely $m = -\frac {4}{3}$ and $\frac {4}{3}$

but i don't know how to tackle b), could someone show me please? thankyou
I assume that you have calculated the coordinates of the points of intersection. Then you've got:

$x=\dfrac{5m \pm\sqrt{9m^2-16}}{1+m^2}$

Plug in the values for m to get the x-coordinate of A or B. Plug in this x-value into the equation of the tangent to get the y-coordinate. Use the distance formula to calculate the length.

3. Well, the obvious way would be to find the points A and B and use the distance formula! And that's easy.

I presume that you found 4/3 and -4/3 by replacing y in [tex]x^2+ y^2- 10y+ 16= 0[tex] by mx to get $(m^2+ 1)x^2- 10mx+ 16= 0$ and arguing that, since a tangent line touches the circle in only one place, that equation must have a single solution for x and so it discriminant, $b^2- 4ac= (-10m)^2- 4(16)(1+ m^2)= 0$. Solving that for m gave you the two solutions.

And, since the discrimant is 0, the quadratic formula reduces to $x= -\frac{b}{2a}= \frac{5m}{1+ m^2}$. Put m equal to 4/3 and -4/3 in that to find the two values of x. Of course, y is just mx.

And, in fact, I note that the center of the circle is on the y-axis so the y values of points A and B are the same! The distance from A to B is just the difference of the x coordinates. How simple is that?

4. ok i've used the simplified quadratic formula (because the discriminant is 0) $\frac{5m}{1+ m^2}$ then $\frac{5(4/3)}{1 + (16/9)}$ which came to $\frac {60/9}{25/9}$ and i've arrived at x = 2.4 and y = 3.2.
then using the distance formula like so $\sqrt {2.4^2 + 3.2^2}$ which come to $\sqrt {5.76 + 10.24}$ which equals $\sqrt {16}$ which is 4. the answer given to me by the book is 4.8 though? what have i done wrong here?

5. Originally Posted by mark
ok i've used the simplified quadratic formula (because the discriminant is 0) $\frac{5m}{1+ m^2}$ then $\frac{5(4/3)}{1 + (16/9)}$ which came to $\frac {60/9}{25/9}$ and i've arrived at x = 2.4 and y = 3.2. <<<<<<<<< That means the point A is A(2.4, 3.2)

Now use $\color{blue}\bold{m=-\dfrac43}$ to get the coordinates of B(-2.4, 3.2)
then using the distance formula like so $\sqrt {2.4^2 + 3.2^2}$ which come to $\sqrt {5.76 + 10.24}$ which equals $\sqrt {16}$ which is 4. the answer given to me by the book is 4.8 though? what have i done wrong here?
ok i've used the simplified quadratic formula (because the discriminant is 0) $\frac{5m}{1+ m^2}$ then $\frac{5(4/3)}{1 + (16/9)}$ which came to $\frac {60/9}{25/9}$ and i've arrived at x = 2.4 and y = 3.2.
then using the distance formula like so $\sqrt {2.4^2 + 3.2^2}$ which come to $\sqrt {5.76 + 10.24}$ which equals $\sqrt {16}$ which is 4. the answer given to me by the book is 4.8 though? what have i done wrong here?