Get expert advice and insights on any topic with IDNLearn.com. Discover reliable and timely information on any topic from our network of experienced professionals.
Sagot :
To solve this question, we need to understand the relationship between the slopes of the given lines. A tangent line to a circle at a point [tex]\( Q \)[/tex] is perpendicular to the line passing through the center of the circle and the point [tex]\( Q \)[/tex]. In this question, the equation of the diameter passing through point [tex]\( Q \)[/tex] is given as [tex]\( y = 4x + 2 \)[/tex].
1. Identify the slope of the given line:
The given line [tex]\( y = 4x + 2 \)[/tex] is in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Here, the slope [tex]\( m \)[/tex] of the given line is [tex]\( 4 \)[/tex].
2. Find the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. So, we need to find the negative reciprocal of [tex]\( 4 \)[/tex].
The negative reciprocal of [tex]\( 4 \)[/tex] is calculated as:
[tex]\[ \text{slope of the perpendicular line} = -\frac{1}{4} \][/tex]
3. Determine the slope of the tangent line:
Since the tangent line at point [tex]\( Q \)[/tex] on the circle [tex]\( P \)[/tex] is perpendicular to the line passing through the center and the same point [tex]\( Q \)[/tex], the slope of the tangent line must be:
[tex]\[ -\frac{1}{4} \][/tex]
Therefore, the correct statement describing the slope of the tangent line to circle [tex]\( P \)[/tex] at point [tex]\( Q \)[/tex] is:
C. The slope of the tangent line is [tex]\(-\frac{1}{4}\)[/tex].
1. Identify the slope of the given line:
The given line [tex]\( y = 4x + 2 \)[/tex] is in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Here, the slope [tex]\( m \)[/tex] of the given line is [tex]\( 4 \)[/tex].
2. Find the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. So, we need to find the negative reciprocal of [tex]\( 4 \)[/tex].
The negative reciprocal of [tex]\( 4 \)[/tex] is calculated as:
[tex]\[ \text{slope of the perpendicular line} = -\frac{1}{4} \][/tex]
3. Determine the slope of the tangent line:
Since the tangent line at point [tex]\( Q \)[/tex] on the circle [tex]\( P \)[/tex] is perpendicular to the line passing through the center and the same point [tex]\( Q \)[/tex], the slope of the tangent line must be:
[tex]\[ -\frac{1}{4} \][/tex]
Therefore, the correct statement describing the slope of the tangent line to circle [tex]\( P \)[/tex] at point [tex]\( Q \)[/tex] is:
C. The slope of the tangent line is [tex]\(-\frac{1}{4}\)[/tex].
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. For dependable and accurate answers, visit IDNLearn.com. Thanks for visiting, and see you next time for more helpful information.