Straight Lines and Circles

Equations of Tangents and Normals to a Circle

When you are asked to find the equation of either a tangent (perpendicular to the radius) or a normal (parallel to the radius) of a circle at a certain point, there are a few steps that you have to go through:

1. Figure out the gradient of the radius at the point by calculating the change in the y-coordinates divided by the change in the x-coordinates
2. If you are asked to find the normal, the gradient stays the same, but if you are asked to find the tangent, then take the negative reciprocal of the gradient of the radius (-1/m)
3. Then, using the y-y₁=m(x-x₁) method, see here for more, to get the y=mx+c form

Straight Lines and Circles Intersecting

Sometimes you will be asked to tell if a line and a circle will intersect, for example y=2x+1 and x²+2x+y²-3y+4=5. What you do is substitute the y's in the circle with 2x+1, getting x²+2x+(2x+1)²-3(2x+1)+4 = 5 => x²+2x+4x²+4x+1-6x-3+4 = 5 => 5x²+2=5 => 5x²-3 =0. Then, we can use the discriminant to to figure out if it does intersect. These are what could happen:

• If the discriminant is positive, then it intersects the circle twice
• If the discriminant is negative, then it doesn't intersect the circle at all
• If the discriminat is 0, then it intersects only once, which means that it is a tangent!

If it does intersect, then the resulting quadratic can be solved to get the x-coordinate(s), which can then be put back into the y=2x+1 equation to get the y-coordinate(s).

Straight Lines and Quadratics

Finding out where a line and a quadratic intersects is similar to the line and circle process, including the discriminant and intersection(s) point, but instead of substituing they can be set equal to one another, i.e mx+c=ax²+bx+c