A line that intersects with a circle

[Banks]

Hi,

I'm a bit stumped on a maths question;

Say you're given a co-ordinate for a point on a circle: Point P: (-1, 3). You're told that the line y = -7x - 4 passes through this point, and another point on the circle (point R) and asked to find the coord of point R.

So I substituted the value for y (y = -7x - 4) into the equation for the circle, which is (x - 2)^2 + (y - 7)^2 = 25. However, I keep getting a bit "lost" when expanding and trying to find the value for x. I have struggled a bit with expanding brackets and become a bit overwhelmed with all the terms.

Could someone kindly provide the simplest or most efficient way to do this?

Many thanks

 

[Harry_the_cat]

Hi,

I'm a bit stumped on a maths question;

Say you're given a co-ordinate for a point on a circle: Point P: (-1, 3). You're told that the line y = -7x - 4 passes through this point, and another point on the circle (point R) and asked to find the coord of point R.

So I substituted the value for y (y = -7x - 4) into the equation for the circle, which is (x - 2)^2 + (y - 7)^2 = 25. However, I keep getting a bit "lost" when expanding and trying to find the value for x. I have struggled a bit with expanding brackets and become a bit overwhelmed with all the terms.

Could someone kindly provide the simplest or most efficient way to do this?

Many thanks

Your proposed method is correct. Show what you have done so far and we'll point you in the right direction.

 

[Banks]

Sure, so here's what I did:

(x - 2)^2 + (y - 7)^2 = 25

Substitute y

(x - 2)^2 + (-7x - 4 - 7)^2 = 25 ... becomes ... (x - 2)^2 + (-7x - 11)^2 = 25

Expand and add like terms

x^2 - 199x + 125 = 25

Subtract 25

x^2 - 199x + 100 = 0

So at this point I'm trying to get x on one side but don't know the simplest way to do it. Do I subtract 100 from each side, then divide by -199?
It doesn't feel like I'm doing it correctly.

Thanks

 

[Deleted member 4993]

Sure, so here's what I did:

(x - 2)^2 + (y - 7)^2 = 25

Substitute y

(x - 2)^2 + (-7x - 4 - 7)^2 = 25 ... becomes ... (x - 2)^2 + (-7x - 11)^2 = 25

Expand and add like terms

x^2 - 199x + 125 = 25 ← This is incorrect. Please show your expanded equation.

Subtract 25

x^2 - 199x + 100 = 0 ← This type of equation is solved using quadratic formula. Have you studied "quadratic equation"?

So at this point I'm trying to get x on one side but don't know the simplest way to do it. Do I subtract 100 from each side, then divide by -199?
It doesn't feel like I'm doing it correctly.

Thanks

.

 

[Banks]

Aha, I must've made a careless mistake - not entirely sure how.

So I re-did it for the 100th time... and got 50x^2 + 150x + 100 = 0. I think that I didn't add 49x^2 to x^2.

Then I factored it and got what I wanted: The co-ords of point R which are (-2, 10).

Thanks for answering guys.