Hard maths question

Anyone know how to solve this? I found it in this paper: http://www.m4ths.com/uploads/3/2/7/4/3274186/358.pdf

Using algebra, show that part of the line 3x + 4y = 0 is a diameter of the circle
with equation x^2 + y^2 = 25.

So far, I managed to convert 3x + 4y = 0 into y = -3/4x + 0... and that's pretty much it, lol.

If a segment of that line is the diameter then the line goes through the centre of the circle

But it says using algebra, and I'm pretty confused if we're allowed to show it through a diagram or not. Thanks for the reply btw

Yes, you use algebra to show it analytically.

Substitute $y=-\frac{3}{4}x$ into the circle and solve for $x$. Then find the coordinates of intersection between the line and the circle, and then show the distance between these two points is twice the radius.

This is far simpler.

You already know the circle's centre, $(0,0)$ using the equation, just show that the line passes through it using a substitution.

That's what I would've suggested in the first place but that doesn't involve algebra work as the question asks.

It is completely algebraic - you know the centre from the circle's equation: $(x-a)^{2}+(y-b)^{2}=r^{2}$, where $(a,b)$ is your circle's centre.

Also, there is no need to find a diameter's length.

I doubt that's what they're looking for, one liners aren't worth 6 marks and a full page.

I doubt that's what they're looking for, one liners aren't worth 6 marks and a full page.

I really do agree, but I don't see what else you need to add.

Edit2: Edit: I realise the paper is a GCSE level test, of which the circle equation won't be part of the syllabus. no, the circle equation is on the syllabus, i'm really confused about why it's 6mks haha