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Currently I am looking at a graph of a circle. The diameter is y=2x+3 Tangent at point E cuts the x-axis at F (12;0) 1. find the coordinates of E 2. find the coordinates of G and H (H being the centre)

You'll find the image useful! enter image description here

Nicole Carr
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  • There is an overflow of questions from you... without apparent work. Try first, then say us what you have tried... – Jean Marie Apr 14 '16 at 16:03
  • Say us? If I knew how to do the work then trust me I would! I have not looked at graphs like this in years and can't find anything that genuinely helps me figure it out. – Nicole Carr Apr 14 '16 at 16:11
  • All right: what do you mean when you say "the diameter is $y=2x+3$" ? – Jean Marie Apr 14 '16 at 16:22
  • that is the equation of the line known as the diameter. – Nicole Carr Apr 14 '16 at 16:33
  • Some information is missing. Is the centre of the circle assumed to lie on the $y$-axis? – grand_chat Apr 14 '16 at 16:40
  • I had a look at your figure. Begin by drawing the line with equation $y=2x+3$ in a correct way: it is completely false and you cannot do geometry with completely erroneous figures. For having a precise drawing, use squared paper. – Jean Marie Apr 14 '16 at 16:41
  • Thank you, however it was given to me by a teacher (this is just a rough drawing showing the shape) I have all the co-ordinates so the drawing does not matter. – Nicole Carr Apr 14 '16 at 16:49
  • grand_chat, no, the middle of the circle is point H and the O that lies on the y axis is the origin. – Nicole Carr Apr 14 '16 at 16:50

1 Answers1

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Hints: The diameter lies on the line $y=2x+3$, which has a slope of $2$, so the tangent line has a slope of $-\frac12$. The tangent line contains the point $(12,0)$, so using the point-slope form we get $$ y-0=-\frac12(x-12) $$ as the equation of the tangent line.

(1) The point $E$ is the intersection of the tangent line with the diameter line.

(2) $G$ is the $y$-intercept of the tangent line.

(3) Suppose the centre $H$ of the circle has coordinates $(a,b)$. The centre $H$ lies on the diameter line, so $a$ and $b$ satisfy $$ b=2a+3.$$ That's one equation for $a$ and $b$. But additional information is needed to completely determine the values of $a$ and $b$. For example, it looks like $H$ lies on the $y$-axis; if so, then this implies that $a=0$, and you can solve for $b$. Without additional information you can imagine infinitely many possibilities for the location of $H$.

(4) The shape $OFEH$ can be viewed as the difference of two triangles, $OFG$ and $HEG$.

grand_chat
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