Answer :
Let's solve the given questions step-by-step:
### 4.1.1 Determine the coordinates of A, the centre of the circle and the length of the radius, r.
Let’s assume the center of the circle, [tex]\( A(x, y) \)[/tex], is [tex]\( (a, b) \)[/tex] and it has a radius [tex]\( r \)[/tex].
### 4.1.2 Calculate the value of p if [tex]\( N(1, p) \)[/tex] with [tex]\( p > 0 \)[/tex] is a point on the circle.
Since point [tex]\( N(1, p) \)[/tex] lies on the circle, it must satisfy the equation of the circle. Using the distance formula between points [tex]\( (a, b) \)[/tex] and [tex]\( (1, p) \)[/tex]:
[tex]\[ \sqrt{(a - 1)^2 + (b - p)^2} = r \][/tex]
Squaring both sides:
[tex]\[ (a - 1)^2 + (b - p)^2 = r^2 \][/tex]
To find [tex]\( p \)[/tex], we need the equation of the circle, but since we don’t have it, the above relationship is the best general result we can obtain. If specific values for [tex]\( a, b, r \)[/tex] were provided, we could solve for [tex]\( p \)[/tex].
### 4.1.3 Determine the equation of the tangent to the circle at [tex]\( N \)[/tex].
The tangent line to a circle at a point [tex]\( (x_1, y_1) \)[/tex] on the circle [tex]\( (x - a)^2 + (y - b)^2 = r^2 \)[/tex] can be found using the formula:
[tex]\[ (x_1 - a)(x - a) + (y_1 - b)(y - b) = r^2 \][/tex]
For point [tex]\( N(1, p) \)[/tex]:
[tex]\[ (1 - a)(x - a) + (p - b)(y - b) = r^2 \][/tex]
Solving this for the tangent line in [tex]\( x \)[/tex] and [tex]\( y \)[/tex] terms, you will get the equation of the tangent line at [tex]\(N(1, p)\)[/tex].
### Determine the values of k for which point A will be inside circle B.
The second circle is centered at [tex]\( B(4, 0) \)[/tex] with equation [tex]\( (x - 4)^2 + y^2 = k^2 \)[/tex].
For point [tex]\( A(a, b) \)[/tex] to lie inside circle [tex]\( B \)[/tex], the distance between [tex]\( A \)[/tex] and [tex]\( B \)[/tex] must be less than the radius [tex]\( k \)[/tex] of circle [tex]\( B \)[/tex]:
[tex]\[ \sqrt{(a - 4)^2 + (b - 0)^2} < k \][/tex]
Squaring both sides:
[tex]\[ (a - 4)^2 + b^2 < k^2 \][/tex]
Thus, [tex]\( k \)[/tex] must be greater than the distance between the centers of the circles [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ k > \sqrt{(a - 4)^2 + b^2} \][/tex]
In the general case, without specific coordinates for [tex]\( a \)[/tex] and [tex]\( b \)[/tex], this is the most precise form of the equation. If specific values for [tex]\( a \)[/tex] and [tex]\( b \)[/tex] were known, [tex]\( k \)[/tex] could be explicitly calculated.
### Summary:
- The center of the circle [tex]\( A \)[/tex] is given as [tex]\( (a, b) \)[/tex] and radius [tex]\( r \)[/tex].
- The value of [tex]\( p \)[/tex] is determined by substituting [tex]\( N(1, p) \)[/tex] into the circle's equation.
- The tangent to the circle at [tex]\( N \)[/tex] can be derived from the general tangent equation provided.
- The value of [tex]\( k \)[/tex] is determined by ensuring that the distance between circle [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is less than [tex]\( k \)[/tex].
Further solution refinement requires knowing the exact values for [tex]\( a, b, r \)[/tex].
### 4.1.1 Determine the coordinates of A, the centre of the circle and the length of the radius, r.
Let’s assume the center of the circle, [tex]\( A(x, y) \)[/tex], is [tex]\( (a, b) \)[/tex] and it has a radius [tex]\( r \)[/tex].
### 4.1.2 Calculate the value of p if [tex]\( N(1, p) \)[/tex] with [tex]\( p > 0 \)[/tex] is a point on the circle.
Since point [tex]\( N(1, p) \)[/tex] lies on the circle, it must satisfy the equation of the circle. Using the distance formula between points [tex]\( (a, b) \)[/tex] and [tex]\( (1, p) \)[/tex]:
[tex]\[ \sqrt{(a - 1)^2 + (b - p)^2} = r \][/tex]
Squaring both sides:
[tex]\[ (a - 1)^2 + (b - p)^2 = r^2 \][/tex]
To find [tex]\( p \)[/tex], we need the equation of the circle, but since we don’t have it, the above relationship is the best general result we can obtain. If specific values for [tex]\( a, b, r \)[/tex] were provided, we could solve for [tex]\( p \)[/tex].
### 4.1.3 Determine the equation of the tangent to the circle at [tex]\( N \)[/tex].
The tangent line to a circle at a point [tex]\( (x_1, y_1) \)[/tex] on the circle [tex]\( (x - a)^2 + (y - b)^2 = r^2 \)[/tex] can be found using the formula:
[tex]\[ (x_1 - a)(x - a) + (y_1 - b)(y - b) = r^2 \][/tex]
For point [tex]\( N(1, p) \)[/tex]:
[tex]\[ (1 - a)(x - a) + (p - b)(y - b) = r^2 \][/tex]
Solving this for the tangent line in [tex]\( x \)[/tex] and [tex]\( y \)[/tex] terms, you will get the equation of the tangent line at [tex]\(N(1, p)\)[/tex].
### Determine the values of k for which point A will be inside circle B.
The second circle is centered at [tex]\( B(4, 0) \)[/tex] with equation [tex]\( (x - 4)^2 + y^2 = k^2 \)[/tex].
For point [tex]\( A(a, b) \)[/tex] to lie inside circle [tex]\( B \)[/tex], the distance between [tex]\( A \)[/tex] and [tex]\( B \)[/tex] must be less than the radius [tex]\( k \)[/tex] of circle [tex]\( B \)[/tex]:
[tex]\[ \sqrt{(a - 4)^2 + (b - 0)^2} < k \][/tex]
Squaring both sides:
[tex]\[ (a - 4)^2 + b^2 < k^2 \][/tex]
Thus, [tex]\( k \)[/tex] must be greater than the distance between the centers of the circles [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ k > \sqrt{(a - 4)^2 + b^2} \][/tex]
In the general case, without specific coordinates for [tex]\( a \)[/tex] and [tex]\( b \)[/tex], this is the most precise form of the equation. If specific values for [tex]\( a \)[/tex] and [tex]\( b \)[/tex] were known, [tex]\( k \)[/tex] could be explicitly calculated.
### Summary:
- The center of the circle [tex]\( A \)[/tex] is given as [tex]\( (a, b) \)[/tex] and radius [tex]\( r \)[/tex].
- The value of [tex]\( p \)[/tex] is determined by substituting [tex]\( N(1, p) \)[/tex] into the circle's equation.
- The tangent to the circle at [tex]\( N \)[/tex] can be derived from the general tangent equation provided.
- The value of [tex]\( k \)[/tex] is determined by ensuring that the distance between circle [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is less than [tex]\( k \)[/tex].
Further solution refinement requires knowing the exact values for [tex]\( a, b, r \)[/tex].
