My Math Forum Tangents to Circle

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February 3rd, 2018, 10:53 PM   #1
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Tangents to Circle

Hi all

Can someone help me to find the co-ordinates of point P please?

I understand that the tangents are the same length (because they're both coming from a single external point).

Does this mean an isosceles triangle is involved in solving it?

Any pointers on the next step would be appreciated.

Thanks

Jim
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Last edited by skipjack; February 4th, 2018 at 02:12 PM.

 February 3rd, 2018, 11:06 PM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,122 Thanks: 1102 It should be pretty clear that i) $B=(0,-5)$ ii) $P=(P_x,-5)$ so we just need to solve for $P_x$ The tangent line at $(3,4)$ will have slope $-\dfrac{3}{4}$ (show this) The slope of this line is also $\dfrac{4-(-5)}{3-P_x}= \dfrac{9}{3-P_x}$ $\dfrac{9}{3-P_x} = -\dfrac{3}{4}$ $P_x = 15$
 February 3rd, 2018, 11:41 PM #3 Newbie   Joined: Dec 2014 From: UK Posts: 11 Thanks: 0 Hi Romsek That is a brilliant answer which I understand. The only part that I don't understand is how you work out the gradient of the tangent? What is the simplest way of doing that please? Regards Jim
February 4th, 2018, 12:08 AM   #4
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Quote:
 Originally Posted by Jimbo77 Hi Romsek That is a brilliant answer which I understand. The only part that I don't understand is how you work out the gradient of the tangent? What is the simplest way of doing that please? Regards Jim
What do you know about the tangent line of a circle with respect to the radius of the circle? Use this.

 February 4th, 2018, 02:14 AM #5 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,879 Thanks: 1087 Math Focus: Elementary mathematics and beyond Or use the distance formula and the fact that the tangents are of equal length. $$(x-3)^2+81=x^2\implies x=15$$ Thanks from romsek
February 4th, 2018, 06:10 AM   #6
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Quote:
 Originally Posted by Micrm@ss What do you know about the tangent line of a circle with respect to the radius of the circle? Use this.
I know it is always at right angles? Is this correct?

February 4th, 2018, 06:13 AM   #7
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Quote:
 Originally Posted by greg1313 Or use the distance formula and the fact that the tangents are of equal length. $$(x-3)^2+81=x^2\implies x=15$$
Sorry could you explain this a bit more please?

 February 4th, 2018, 11:48 AM #8 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,879 Thanks: 1087 Math Focus: Elementary mathematics and beyond
 February 4th, 2018, 11:59 AM #9 Newbie   Joined: Dec 2014 From: UK Posts: 11 Thanks: 0 Thanks

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