Could anyone help me understand for Q11b on the 2022 AS Paper 1 (Edexcel Maths) why the perpendicular line passing through line l and the centre of the circle C is the shortest distance between the circle and the line l?

Question:

Figure 3 shows the circle C with equation

x^2 + y^2 – 10x – 8y + 32 = 0

and the line l with equation

2y + x + 6 = 0

(a) Find

(i) the coordinates of the centre of C,

(ii) the radius of C.

(b) Find the shortest distance between C and l.

(5)

Link to the paper (see Q11b) https://www.physicsandmathstutor.com/pdf-pages/?pdf=https%3A%2F%2Fpmt.physicsandmathstutor.com%2Fdownload%2FMaths%2FA-level%2FPapers%2FEdexcel%2FAS-Paper-1%2FQP%2FJune%25202022%2520QP.pdf

Question:

Figure 3 shows the circle C with equation

x^2 + y^2 – 10x – 8y + 32 = 0

and the line l with equation

2y + x + 6 = 0

(a) Find

(i) the coordinates of the centre of C,

(ii) the radius of C.

(b) Find the shortest distance between C and l.

(5)

Link to the paper (see Q11b) https://www.physicsandmathstutor.com/pdf-pages/?pdf=https%3A%2F%2Fpmt.physicsandmathstutor.com%2Fdownload%2FMaths%2FA-level%2FPapers%2FEdexcel%2FAS-Paper-1%2FQP%2FJune%25202022%2520QP.pdf

(edited 11 months ago)

haha, i got this nasty question on my test too. when you look at the diagram, the shortest distance connecting the edge of the circle and line l would be the normal(perpendicular line). To find that distance, you can only use the centre of the circle as a reference point, because when the normal is connected to the centre, it touches the edge of the circle, which is the shortest distance. by finding that distance and taking away the radius (which is 3 in this case), you get the shortest distance! hope this helps

Original post by swaggiekaty

haha, i got this nasty question on my test too. when you look at the diagram, the shortest distance connecting the edge of the circle and line l would be the normal(perpendicular line). To find that distance, you can only use the centre of the circle as a reference point, because when the normal is connected to the centre, it touches the edge of the circle, which is the shortest distance. by finding that distance and taking away the radius (which is 3 in this case), you get the shortest distance! hope this helps

thats incredibly helpful, thank you!

Original post by yzven

thats incredibly helpful, thank you!

thanks for this

with the centre of the circle how do you work the points on the tangent

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