IDNLearn.com: Your one-stop destination for reliable answers to diverse questions. Our experts provide timely and accurate responses to help you navigate any topic or issue with confidence.
Sagot :
Let's solve this problem step by step.
### Part (a): Show that the line does not intersect the circle.
We start by analyzing the equations of the line and the circle:
- The line is given by [tex]\( y = x - 5 \)[/tex].
- The circle is given by [tex]\( (x - 3)^2 + (y - 4)^2 = 4 \)[/tex].
To check if they intersect, we substitute the equation of the line into the equation of the circle. Substitute [tex]\( y \)[/tex] from the line equation into the circle equation:
[tex]\[ (x - 3)^2 + ((x - 5) - 4)^2 = 4 \][/tex]
Simplifying the equation, we have:
[tex]\[ (x - 3)^2 + (x - 9)^2 = 4 \][/tex]
Calculating the squares:
[tex]\[ (x - 3)^2 + (x - 9)^2 = (x^2 - 6x + 9) + (x^2 - 18x + 81) \][/tex]
Combine like terms:
[tex]\[ 2x^2 - 24x + 90 = 4 \][/tex]
Subtract 4 from both sides:
[tex]\[ 2x^2 - 24x + 86 = 0 \][/tex]
Simplify by dividing everything by 2:
[tex]\[ x^2 - 12x + 43 = 0 \][/tex]
To determine if real solutions exist, calculate the discriminant [tex]\( \Delta \)[/tex]:
[tex]\[ \Delta = b^2 - 4ac = (-12)^2 - 4(1)(43) = 144 - 172 = -28 \][/tex]
Since the discriminant is negative ([tex]\(-28\)[/tex]), the quadratic equation [tex]\( x^2 - 12x + 43 = 0 \)[/tex] has no real solutions. Therefore, the line and the circle do not intersect.
### Part (b): Find the coordinates of [tex]\( P \)[/tex] and the exact value of the shortest distance from [tex]\( P \)[/tex] to the circle.
To find the point [tex]\( P \)[/tex] on the line [tex]\( y = x - 5 \)[/tex] that is closest to the center of the circle (3, 4), we minimize the distance between a point on the line and the center of the circle.
Consider a point [tex]\( P \)[/tex] on the line given by the coordinates [tex]\( (x, y) \)[/tex] where [tex]\( y = x - 5 \)[/tex]. Let [tex]\( P = (x, x - 5) \)[/tex].
The distance [tex]\( D \)[/tex] between the point [tex]\( P \)[/tex] and the center of the circle [tex]\( (3, 4) \)[/tex] is given by the distance formula:
[tex]\[ D = \sqrt{(x - 3)^2 + ((x - 5) - 4)^2} \][/tex]
Simplify the expression inside the square root:
[tex]\[ D = \sqrt{(x - 3)^2 + (x - 9)^2} \][/tex]
To minimize this distance, we can consider the derivative. However, we can also directly use symmetry and geometric considerations. Since the equation of the line is linear and we're calculating a distance to a fixed point, the closest [tex]\( x \)[/tex] will be the perpendicular projection.
Minimize the distance, recognize that it happens when the derivative of [tex]\( D \)[/tex] with respect to [tex]\( x \)[/tex] is zero, find the critical points.
Solving the differentiation results:
[tex]\[ P = (6, 1) \][/tex]
Finally, calculate the exact distance from [tex]\( P = (6, 1) \)[/tex] to the center of the circle [tex]\( (3, 4) \)[/tex]:
[tex]\[ D = \sqrt{(6 - 3)^2 + (1 - 4)^2} = \sqrt{3^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \][/tex]
### Summary of Results:
(a) The line does not intersect the circle.
(b) The coordinates of [tex]\( P \)[/tex] are [tex]\( (6, 1) \)[/tex] and the exact value of the shortest distance from [tex]\( P \)[/tex] to the circle is [tex]\( 3\sqrt{2} \)[/tex].
### Part (a): Show that the line does not intersect the circle.
We start by analyzing the equations of the line and the circle:
- The line is given by [tex]\( y = x - 5 \)[/tex].
- The circle is given by [tex]\( (x - 3)^2 + (y - 4)^2 = 4 \)[/tex].
To check if they intersect, we substitute the equation of the line into the equation of the circle. Substitute [tex]\( y \)[/tex] from the line equation into the circle equation:
[tex]\[ (x - 3)^2 + ((x - 5) - 4)^2 = 4 \][/tex]
Simplifying the equation, we have:
[tex]\[ (x - 3)^2 + (x - 9)^2 = 4 \][/tex]
Calculating the squares:
[tex]\[ (x - 3)^2 + (x - 9)^2 = (x^2 - 6x + 9) + (x^2 - 18x + 81) \][/tex]
Combine like terms:
[tex]\[ 2x^2 - 24x + 90 = 4 \][/tex]
Subtract 4 from both sides:
[tex]\[ 2x^2 - 24x + 86 = 0 \][/tex]
Simplify by dividing everything by 2:
[tex]\[ x^2 - 12x + 43 = 0 \][/tex]
To determine if real solutions exist, calculate the discriminant [tex]\( \Delta \)[/tex]:
[tex]\[ \Delta = b^2 - 4ac = (-12)^2 - 4(1)(43) = 144 - 172 = -28 \][/tex]
Since the discriminant is negative ([tex]\(-28\)[/tex]), the quadratic equation [tex]\( x^2 - 12x + 43 = 0 \)[/tex] has no real solutions. Therefore, the line and the circle do not intersect.
### Part (b): Find the coordinates of [tex]\( P \)[/tex] and the exact value of the shortest distance from [tex]\( P \)[/tex] to the circle.
To find the point [tex]\( P \)[/tex] on the line [tex]\( y = x - 5 \)[/tex] that is closest to the center of the circle (3, 4), we minimize the distance between a point on the line and the center of the circle.
Consider a point [tex]\( P \)[/tex] on the line given by the coordinates [tex]\( (x, y) \)[/tex] where [tex]\( y = x - 5 \)[/tex]. Let [tex]\( P = (x, x - 5) \)[/tex].
The distance [tex]\( D \)[/tex] between the point [tex]\( P \)[/tex] and the center of the circle [tex]\( (3, 4) \)[/tex] is given by the distance formula:
[tex]\[ D = \sqrt{(x - 3)^2 + ((x - 5) - 4)^2} \][/tex]
Simplify the expression inside the square root:
[tex]\[ D = \sqrt{(x - 3)^2 + (x - 9)^2} \][/tex]
To minimize this distance, we can consider the derivative. However, we can also directly use symmetry and geometric considerations. Since the equation of the line is linear and we're calculating a distance to a fixed point, the closest [tex]\( x \)[/tex] will be the perpendicular projection.
Minimize the distance, recognize that it happens when the derivative of [tex]\( D \)[/tex] with respect to [tex]\( x \)[/tex] is zero, find the critical points.
Solving the differentiation results:
[tex]\[ P = (6, 1) \][/tex]
Finally, calculate the exact distance from [tex]\( P = (6, 1) \)[/tex] to the center of the circle [tex]\( (3, 4) \)[/tex]:
[tex]\[ D = \sqrt{(6 - 3)^2 + (1 - 4)^2} = \sqrt{3^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \][/tex]
### Summary of Results:
(a) The line does not intersect the circle.
(b) The coordinates of [tex]\( P \)[/tex] are [tex]\( (6, 1) \)[/tex] and the exact value of the shortest distance from [tex]\( P \)[/tex] to the circle is [tex]\( 3\sqrt{2} \)[/tex].
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. IDNLearn.com has the solutions to your questions. Thanks for stopping by, and come back for more insightful information.
