Maths A Level Coordinate Geometry Question

Hiya I'm pretty stuck on this question:

The tangent to the circle x^2 + (y+4)^2 = 25 at the point (-4, -1) intersects the x-axis at A and the y-axis at B. Find the exact area of the triangle AOB.

I know that the centre of the circle is (0, -4) and its radius is 5 units, and I tried drawing it out to try and visualise it and find the gradient of the tangent or something, but I'm a bit unsure as to what to do.

Any help would be appreciated, thanks.

Reply 1

mqb2766

21

Original post by aditi_idk

Hiya I'm pretty stuck on this question:

The tangent to the circle x^2 + (y+4)^2 = 25 at the point (-4, -1) intersects the x-axis at A and the y-axis at B. Find the exact area of the triangle AOB.

I know that the centre of the circle is (0, -4) and its radius is 5 units, and I tried drawing it out to try and visualise it and find the gradient of the tangent or something, but I'm a bit unsure as to what to do.

Any help would be appreciated, thanks.

The straightforward way is to find the equation of the tangent, then find the points of intersection with the axes and hence the area.

Reply 2

Bookworm524

19

I'm slightly confused how there could be a tangent at (-4,1) when the point is inside the circle. Are you sure these are the numbers in the question?

(edited 11 months ago)

Reply 3

aditi_idk

OP

10

Original post by Bookworm524

I'm slightly confused how there could be a tangent at (-4,1) when the point is inside the circle. Are you sure these are the numbers in the question?

I believe the point is (-4,-1), not (-4, 1)

Reply 4

bigplongus

15

Original post by Bookworm524

I'm slightly confused how there could be a tangent at (-4,1) when the point is inside the circle. Are you sure these are the numbers in the question?

(-4,-1)

image_2023-05-25_133515897.png74.4KB

Reply 5

bigplongus

15

Original post by aditi_idk

Hiya I'm pretty stuck on this question:

The tangent to the circle x^2 + (y+4)^2 = 25 at the point (-4, -1) intersects the x-axis at A and the y-axis at B. Find the exact area of the triangle AOB.

I know that the centre of the circle is (0, -4) and its radius is 5 units, and I tried drawing it out to try and visualise it and find the gradient of the tangent or something, but I'm a bit unsure as to what to do.

Any help would be appreciated, thanks.

green line is the tangent, try working it out and not using the values on the graph since you wont have it in an exam, i worked it out and this is what it looks like
the purple line is a hint, if you need elaboration or further explanation just ask

image_2023-05-25_134345834.png148.5KB

Reply 6

BiggestIdiot

3

Original post by aditi_idk

Hiya I'mi pretty stuck on this question:

The tangent to the circle x^2 + (y+4)^2 = 25 at the point (-4, -1) intersects the x-axis at A and the y-axis at B. Find the exact area of the triangle AOB.

I know that the centre of the circle is (0, -4) and its radius is 5 units, and I tried drawing it out to try and visualise it and find the gradient of the tangent or something, but I'm a bit unsure as to what to do.

Any help would be appreciated, thanks.

I got you, in case you still need help, this was a fiendishly difficult question and took me a while to solve, but, alas, after a very long while, I did.

First, what you want to do is find the coordinates of the centre of the circle, which are found by:

For equation: (x+a)^2+(y+b)^2=r^2
Coordinates of centre of circle: (-a,-b)

In this case: (x+0)^2+(y+4)^2=25
Coordinates centre (0,-4)

Step two, find the gradient of the line from (0,-4) to (-4,-1), GCSE stuff, which you should know, also make sure you draw a nice, large detailed diagram to help you follow through.

Gradient from (0,4) to (-4,-1)= 3/-4

Next, make sure to sketch the whole tangent line touching the circle on your diagram at (-4,-1), doesn't have to be to scale, duh.

If you haven't forgotten your GCSEs the moment you walked out, like I did, you'll recall that tangents are perpendicular to the radius line, so next, find the perpendicular gradient, which is 4/3, Google it if you don't know how to find the perpendicular gradient.

Use y=mx+c for the equation of the tangent, so knowing that tangent crosses (-4,-1):

-1=-4 x 4/3 + c
c= 13/3

Therefore point B is (0,13/3) which sucks. Point A is when y=0

So 0= 4/3 x X+ 13/3
X=-13/4

So x-intercept= (-13/4,0), which also sucks. Draw the triangle not to scale and visualise what AOB looks like. By the way, the O means origin, so point O is (0,0), I initially thought it meant the centre of the circle (0,-4) which confused me. Then figure out the height of the triangle using the coordinates I just gave you or that you found to draw a not to scale triangle and multiply its base length by its height which can be figured out using Pythagoras theorem, ez GCSE stuff again and then divide the whole thing by two, which is how you figure out the area of a triangle, duh, you should get 169/24. Hope that helped, it took ma ages to do because I'm pretty dumb but the answer is correct so self esteem went 🔝

Maths A Level Coordinate Geometry Question

Quick Reply

Related discussions

Latest

Trending

Trending

Articles for you