Circle Tangent question

[apple2357]

Any thoughts on this question?
The standard way of doing it would be using the discriminant. So the equation of the line is y=m(x-5) and substituting this into the circle equation:

This needs tidying up:

And then using the discriminant, but the algebra gets horrendous and its meant to be an exam question for 16 year olds.

Is there something i am missing?

 

[skeeter]

I had to use some trig, tangent functions for slopes and the sum/difference identity

 

[R.M.]

Complete the square on the circle equation to put it into standard form. You now can find the coordinates of the center. Call the center C, the tangent points A and B, and the intersection of the tangents D. Now CA is perpendicular to L1 and CB is perpendicular to L2. Creating CD gives you two congruent right triangles. You can find CD from the distance formula; circle equation gives you CA & CB. DA = DB, so you can find that length and then use it to find the coordinates of A and B. Now you can find the slope of each line, and ultimately their equations.

I'm sure there's a more efficient way, but this is the first one that came to mind.

 

[Steven G]

So the equation of the line is y=m(x-5). Which line!!!!

 

[apple2357]

So the equation of the line is y=m(x-5). Which line!!!!

A line passing through (5,0)

 

[apple2357]

Exploring another approach...

 

[Dr.Peterson]

Any thoughts on this question?
The standard way of doing it would be using the discriminant. So the equation of the line is y=m(x-5) and substituting this into the circle equation:

This needs tidying up:

And then using the discriminant, but the algebra gets horrendous and its meant to be an exam question for 16 year olds.

Is there something i am missing?

What you did is just what I tried, on the assumption that this is for an algebra class, so geometry and trig would not be expected. But I stopped just where you did, and came back to see what you had done, because although it can be solved in principle, I agree it's not at all what I'd expect on an exam.

Can you tell us more about its source? What topics were covered?

 

[apple2357]

Yes, it was sample question for a new set of assessments for A level exams which are sat in the UK ( age 16-18). It was given to me by a student who wanted some help and i thought it was straightforward until i started. It is first year material, so it wouldn't involve much more than quadratics /discriminant usually and certainly not calculus of implicit equations or any trig identities.

I think it's a rogue question and was probably prepared in a rush. I think if the question gave us the y intercept instead of the x intercept it would fall out more easily and the algebra would be reasonable. I am convinced of that now, given the calibre here can't see an easy way forward with it.

I just wasn't sure if i was missing something obvious.

 

[Steven G]

A line passing through (5,0)

Exactly my point!! How many line do you see going through (5,0)???

 

[Steven G]

Basically you just want to find the intersection of two circles: (x+2)^2 + (y+6)^2 = 17 and (x-5)^2 + y^2 = 85. Did you get that far??

 

[jonah2.0]

Beer soaked vision follows.

Basically you just want to find the intersection of two circles: (x+2)^2 + (y+6)^2 = 17 and (x-5)^2 + y^2 = 85. Did you get that far??

(X - 3/2)^2 + (y + 3)^2 = 85/4

 

[markraz]

Hi. had a similar problem and with some additional work this helped me:

source: https://math.stackexchange.com/ques...enter-of-circle-tangent-to-two-lines-in-space

 

[jonah2.0]

Beer soaked reckoning follows.

Any thoughts on this question?
The standard way of doing it would be using the discriminant. So the equation of the line is y=m(x-5) and substituting this into the circle equation:

This needs tidying up:

And then using the discriminant, but the algebra gets horrendous and its meant to be an exam question for 16 year olds.

Is there something i am missing?

Yes, it was sample question for a new set of assessments for A level exams which are sat in the UK ( age 16-18). It was given to me by a student who wanted some help and i thought it was straightforward until i started. It is first year material, so it wouldn't involve much more than quadratics /discriminant usually and certainly not calculus of implicit equations or any trig identities.

I think it's a rogue question and was probably prepared in a rush. I think if the question gave us the y intercept instead of the x intercept it would fall out more easily and the algebra would be reasonable. I am convinced of that now, given the calibre here can't see an easy way forward with it.

I just wasn't sure if i was missing something obvious.

I wanted to see how horrendous the algebra really is.
Continuing where you stopped, I get
[MATH]100m^4-240m^3+64m^2+96m+16-100m^4+240m^3-192m^2+240m-92=0[/MATH]which then boils down to
[MATH]-128m^2+336m-76=0[/MATH]Thus, m=1/4 or m=19/8
Tangent lines:
y=(1/4)(x-5)=(1/4)x-5/4
&
y=(19/8)(x-5)=(19/8)x-95/8