Rukasha Maths

GCSE maths Beta

Circle tangents and graph translations · Revision guide

Higher · Algebra · 2 pages in the PDF

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Learning objectives

  • Find a tangent to a circle using a perpendicular radius.
  • Interpret horizontal and vertical graph translations.

Key ideas

A circle centred at the origin has radius given by the square root.

x2+y2=r2x^2+y^2=r^2

For two non-vertical perpendicular lines, gradients are negative reciprocals.

m1m2=−1m_1m_2=-1

A horizontal radius has a vertical tangent; a vertical radius has a horizontal tangent.

The inner subtraction gives a horizontal shift; the outer addition gives a vertical shift.

y=f(x−a)+bshifts by(ab)y=f(x-a)+b\quad\text{shifts by}\quad\binom ab

Worked examples

Example 1

Find the tangent to the circle at the given point.

x2+y2=25,P=(3,4)x^2+y^2=25,\qquad P=(3,4)
32+42=25,mOP=43,mtangent=−343^2+4^2=25,\qquad m_{OP}=\frac43,\quad m_{\text{tangent}}=-\frac34
y−4=−34(x−3)y-4=-\frac34(x-3)
3x+4y=253x+4y=25

Example 2

Find the turning point of the translated quadratic.

y=(x+2)2−3,x+2=0⇒x=−2y=(x+2)^2-3,\qquad x+2=0\Rightarrow x=-2
(−2,−3)(-2,-3)

Watch out

Horizontal and vertical lines need their special tangent cases. Read the horizontal translation with the opposite sign inside the bracket.