Alevel Maths - Circle Geometry- please help :)

1) A circle has equation x^2 + y^2 + 4x + 2y - 12 = 0
A line has the equation x + y = 4
Without solving for points of contact/intersection, determine whether the line intersects the circle, is a tangent to it or does not touch it.

2) The line with equation y = 1/3x + 5 is a tangent to a circle with centre (-2,1). Find the equation of the circle

Please help thankyou x

What are your thoughts?

i’m not too sure. i’ve done the rest of the questions on the paper but the last two i am struggling with.
for the first one i have put the centre into the formula to create the equation for the circle but don’t know where to go from there tbh. I feel as though the second one would involve substitution in some way but again not too sure

i’m not too sure. i’ve done the rest of the questions on the paper but the last two i am struggling with.
for the first one i have put the centre into the formula to create the equation for the circle but don’t know where to go from there tbh. I feel as though the second one would involve substitution in some way but again not too sure

Does a sketch help the first one?

yes thankyou i’ve worked that one out now just the second one i’ve got to do now

Any ideas? We know the gradient of the radius from the centre to where the tangent touches ...

i’m not fully sure would it involve working out a perpendicular gradient of -3 from the 1/3?

For the straight line solve for y/x = and sub it into the circle equation. See how many solutions you get by using b^2-4ac

Is that needed for this question ...

The discriminant can be used to see if there are points of Intersection so yes you can use it to check. As we all know there are many methods that can be used for a-level maths.

Yes but read the question and my hint ...

I’m talking about the first part. This still works you don’t need to know the coordinates and it will determine if it crosses is a tangent or doesn’t touch at all. I’m not quite sure what you’re trying to get at?

There is a massive hint not to do this ... if you find the centre and raduis it's obvious it won't cross that line ...

The hint says do not find the coordinates of intersection. The method I have does not find these coordinates, only the prefixes of the equation. Using the discriminant is not the same as using a quadratic equation, it will give a result of <0 0 or >0 and what that number is determines if it is a tangent or not. As I have said previously, when doing a level maths there are many different ways of coming to the same conclusion. Each way will be marked correct (given you’ve done it correctly. I agree that your method works too I’m saying that they could also have done it this way, and this way will give an immediate answer.

The hint says do not find the coordinates of intersection. The method I have does not find these coordinates, only the prefixes of the equation. Using the discriminant is not the same as using a quadratic equation, it will give a result of <0 0 or >0 and what that number is determines if it is a tangent or not. As I have said previously, when doing a level maths there are many different ways of coming to the same conclusion. Each way will be marked correct (given you’ve done it correctly. I agree that your method works too I’m saying that they could also have done it this way, and this way will give an immediate answer.

The point about not finding solutions is a hint you don't need to find the discriminant - in my view the method I suggested is much nicer and shows you've actually thought about the question and not just ploughed into it. I'd commend any student noticing they don't need to do all that substitution.

Ngl when I saw the question I thought about using the discrimination b^2-4ac so u shouldn’t be shot down for saying that

If that is your view that is completely fine. I’m sure you know that people find different methods easier/harder. The fact that you would commend any student for doing it one way does not matter if both have come to the same conclusion. Coming to a correct answer means the student has understood the question and used their problem solving skills to solve it, no matter what method they choose to use. I’m my view both methods are equally commendable as they show different thinking skills applicable to the question.