Review question

# What's the distance from a point to a line in the $x\text{-}y$ plane? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9702

## Suggestion

Show that the perpendicular distance of the point $(h,k)$ from the straight line $x\cos\alpha+y\sin\alpha=p$ is the numerical value of $h\cos\alpha+k\sin\alpha-p.$ [“Numerical value” means ignoring whether it is positive or negative, that is, the absolute value of this expression.]

Can we draw a diagram that helps here? If we draw the line $x\cos\alpha+y\sin\alpha=p$, what is its gradient? Which angle is $\alpha$ in the diagram?

Calculate the coordinates of the centres of the two circles of radius $5$ which pass through the point $(4,4)$ and touch the straight line $3x-4y-28=0$.

How does this part relate to the first part? Draw a new diagram to get a feel for this.

Can we derive expressions involving the coordinates of the centres of the circles?

Remember that the first part only gives us the “numerical value” of the expression. If you want to find out whether it’s positive or negative, you will have to do some extra work.