Tangents to Circles Flashcards

All 7 cards in this deck

How is the gradient of a tangent related to the gradient of the radius drawn to the point of contact?

e.g. radius of gradient $\frac{5}{2}$

It is the negative reciprocal, because the tangent and radius meet at 90°.

e.g. tangent gradient $-\frac{2}{5}$

What are the steps to find the equation of the tangent to a circle at a given 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. Tangent gradient = negative reciprocal: $-\frac{3}{4}$
3. Use $y - y_1 = m(x - x_1)$: $y - 4 = -\frac{3}{4}(x-3)$

What do you know about the two tangents drawn from the same external point to a circle?

They are equal in length.

What is the size of the angle in a semi-circle?

90°. The angle subtended at any point of the circle by a diameter is a right angle.

What are the steps to find the length of a chord from the radius and the distance of the chord from the centre?

e.g. radius 5, chord 3 from the centre

1. The perpendicular from the centre bisects the chord, giving a right-angled triangle with hypotenuse $r$: sides 3, half-chord, 5
2. Pythagoras for half the chord: $\sqrt{5^2 - 3^2} = 4$
3. Double it: chord $= 8$

What are the steps to test whether a line is a tangent to a circle?

e.g. is $y = 5$ a tangent to $x^2 + y^2 = 25$?

1. Substitute the line into the circle equation: $x^2 + 25 = 25$, i.e. $x^2 = 0$
2. Find the discriminant $b^2 - 4ac$ of the quadratic: $0$
3. Zero = tangent, negative = misses, positive = cuts at two points: tangent

True or false? The perpendicular from the centre of a circle to a chord bisects the chord.

True. It meets the chord at its midpoint, so the radius, half-chord and this perpendicular form a right-angled triangle.