Review question

# Given two lines crossing a circle, can we show these lengths are equal? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9551

## Suggestion

The points $S$, $T$, $U$ and $V$ have coordinates $(s,ms)$, $(t,mt)$, $(u,nu)$ and $(v,nv)$ respectively. The lines $SV$ and $UT$ meet the line $y = 0$ at the points with coordinates $(p,0)$ and $(q,0)$, respectively. Show that $p = \frac{(m - n)sv}{ms - nv},$ and write down a similar expression for $q$.

Could we draw a sketch to keep track of what is going on?

Could we work out the equations of the straight lines involved in this problem?

Given that $S$ and $T$ lie on the circle $x^2 + (y - c)^2 = r^2$, find a quadratic equation satisfied by $s$ and by $t$, and hence determine $st$ and $s + t$ in terms of $m$, $c$ and $r$.

What are the coordinates of $S$?

If we have a quadratic equation whose roots are $s$ and $t$, how are $st$ and $s + t$ related to the coefficients of this equation?