Core 1 Circle question

Hi everyone! I'm stuck on this question:

The line

y= -3x +k

is a tangent to the circle

x^2 + y^2 =10

. Find the possible values of

k

We are not permitted to differentiate the equation of the circle to answer the question (that may be irrelevant anyway).

What I tried:
The line is a tangent, so there is only one solution to the pair of simultaneous equations of the line/circle. Hence, the final factorised form must be

(x+k)^2 = 0

x^2 + 2kx + k^2 = 0

Also, subbing in the equation of the line into the circle equation we get:

x^2 + (-3x+k)^2 = 10

x^2 + 9x^2 -6kx + k^2 - 10 = 0

Then

x^2 + 2kx + k^2 = x^2 + 9x^2 -6kx + k^2 - 10

9x^2 -8kx -10 = 0

But then I have no idea what to do, and I'm not even sure that my method is correct. Please could someone give hints, NOT FULL SOLUTIONS!!

Original post by K-Man_PhysCheM

Hi everyone! I'm stuck on this question:

The line

y= -3x +k

is a tangent to the circle

x^2 + y^2 =10

. Find the possible values of

k

We are not permitted to differentiate the equation of the circle to answer the question (that may be irrelevant anyway).

What I tried:
The line is a tangent, so there is only one solution to the pair of simultaneous equations of the line/circle. Hence, the final factorised form must be

(x+k)^2 = 0

x^2 + 2kx + k^2 = 0

Also, subbing in the equation of the line into the circle equation we get:

x^2 + (-3x+k)^2 = 10

x^2 + 9x^2 -6kx + k^2 - 10 = 0

Then

x^2 + 2kx + k^2 = x^2 + 9x^2 -6kx + k^2 - 10

9x^2 -8kx -10 = 0

But then I have no idea what to do, and I'm not even sure that my method is correct. Please could someone give hints, NOT FULL SOLUTIONS!!

Since the line is tangent to the circle, just plug in the equation of the line into the circle, get the form of

ax^2+bx+c=0

and then the discriminant must be equal to 0 for equal roots - ie tangent.

Reply 2

the bear

20

Original post by K-Man_PhysCheM

x^2 + 9x^2 -6kx + k^2 - 10 = 0

*

form an expression for the discriminant of this, then equate it to zero... *

Reply 3

K-Man_PhysCheM

OP

17

Original post by the bear

form an expression for the discriminant of this, then equate it to zero... *

Wow, can't believe I overlooked that...

Thank you, I can solve it now!

Reply 4

K-Man_PhysCheM

OP

17

Original post by RDKGames

Since the line is tangent to the circle, just plug in the equation of the line into the circle, get the form of

ax^2+bx+c=0

and then the discriminant must be equal to 0 for equal roots - ie tangent.

Thank you! Yeah, I had forgotten about the discriminant. Thank you again.

Reply 5

ljf98

1

so after subbing the equation for Y back into the equation of the circle you should be able to got

x^2 + 9x^2 -6kx + k^2 - 10 = 0

which can now be written as
Ax^2 + Bx^2 + C = 0
10x^2 - (6k)x + (k^2 - 10) = 0

For this to be true the discriminant must be greater than or equal to zero

so B^2 - 4AC >/= 0

This should help find your values of k

Reply 6

the bear

20

Original post by ljf98

so after subbing the equation for Y back into the equation of the circle you should be able to got

x^2 + 9x^2 -6kx + k^2 - 10 = 0

which can now be written as
Ax^2 + Bx^2 + C = 0
10x^2 - (6k)x + (k^2 - 10) = 0

For this to be true the discriminant must be greater than or equal to zero

so B^2 - 4AC >/= 0

This should help find your values of k

they are specifically asking for the tangents so D = 0 not D ≥ 0

Reply 7

K-Man_PhysCheM

OP

17

Original post by ljf98

so after subbing the equation for Y back into the equation of the circle you should be able to got

x^2 + 9x^2 -6kx + k^2 - 10 = 0

which can now be written as
Ax^2 + Bx^2 + C = 0
10x^2 - (6k)x + (k^2 - 10) = 0

For this to be true the discriminant must be greater than or equal to zero

so B^2 - 4AC >/= 0

This should help find your values of k

Thank you, though, as others have pointed out, it should the discriminant is equal to 0 (not greater than or equal to), because the line is tangent to the circle so there is only one solution. Thank you for the response, regardless.