AS Maths Please Help?

Question

Hello,

There are some questions that I am not sure how to tackle:

1) The line 4x + 3y = 24 meets the coordinate axes at A(6, 0) and B(0, 8). The equation of the circle is (x - 3)^2 + (y - 4)^2 = 25. Show that the circle passes through the origin.

2) The line x - 2y - 9 = 0 meets the circle x^2 + (y - 3)^2 = 45 at P. P is (3, -3). The line l is parallel to x - 2y - 9 = 0 and meets the the circle at Q. Find an equation for l.

The answers are apparently:

2) x - 2y + 21 = 0

Thank you in advance.

Answer 1

(1)

The line Equation is 4x + 3y = 24.

Point of Axes are (6,0) and (0,8).

Mid point of (6,0) and (0,8) is the Center of the circle.

h = (x1 + x2)/2

h = (6+0)/2 = 3

k = (y1 + y2)/2

k = (0+8)/2

k = 4.

Therefore the Centre of the circle is (3,4).

Distance from the point (6,0) to the Center of the circle

d² = (6-3)² + (0-4)²

d² = 9 + 16

d² = 25

d = 5

Therefore Radius r = 5.

Equation of the circle is (x - h)² + (y - k)² = r².

(x-3)²+(y-4)² = 25.

Therefore Equation of the Circle is (x-3)²+(y-4)² = 25.

Equation of the Circle Passes through the origin(0,0)

Substitute (0,0) in the Equation of the circle (x-3)²+(y-4)² = 25.

(0-3)²+*(0-4)²=25

25=25

The Equation of the Circle passes through the origin.

Graph

Therefore the Equation of Circle is (x-3)²+(y-4)² = 25 and passes through the origin.

Answer 2

(2)

The Equation of Circle is x²+(y-3)²=25.

Centre of the circle is (h,k) is (0,3) and radius = 5

The line equation is x - 2y = 9 meets circle a point (3,-3).

Now the line equation parallel to the line x - 2y = 9 meets circle at point Q then

Let point Q be (x,y).

Mid point of P and Q is the center of the circle.

h = (x1+x2)/2

0 = (3+x)/2

x = -3

k = (y1+y2)/2

3 = (-3+y)/2

(-3+y) = 6

y = 9

Therefore the point Q is (-3,9)

Equation of the line is x - 2y = 9

2y = x - 9.

y = (1/2)x - (9/2)

Slope of the Equation is m = 1/2

Therefore line equation parallel to the line x - 2y = 9 is

y = mx + b

y = (1/2)x+b

The point Q(-3,9) passes through the line.

9 = (-3/2) + b

b = 9+3/2

b = 21/2

The line equation is y = (1/2)x+21/2

2y = x+21

x - 2y +21 = 0.

Therefore the line equation is x - 2y +21 = 0.