Equation of a Tangent Flashcards

All 4 cards in this deck

What are the steps to find the equation of the tangent to a circle at a point on it?

e.g. tangent to $x^2+y^2=25$ at $(3,4)$

1. Gradient of the radius from the centre to the point: $\frac{4}{3}$
2. Gradient of tangent = negative reciprocal: $-\frac{3}{4}$
3. Substitute into $y - y_1 = m(x - x_1)$: $y - 4 = -\frac{3}{4}(x-3)$, i.e. $3x + 4y = 25$

True or false? A tangent to a circle touches the circle at exactly one point.

True. A line meeting the circle at one point only is a tangent.

A point $(p,1)$ with $p>0$ lies on the circle $x^2+y^2=4$. How do you find $p$ before working out the tangent?

Substitute $y=1$ into the circle equation: $p^2 = 4-1$, so $p = \sqrt{3}$ (take the positive root).

What is the equation of the tangent to $x^2+y^2=r^2$ at the point $(r,0)$?

e.g. $x^2+y^2=9$ at $(3,0)$

The vertical line $x = r$.

e.g. $x = 3$