The line $x + 3y = 17$ is a tangent to a circle at the point $(2, 5)$ - Scottish Highers Maths - Question 15 - 2023
Question 15
The line $x + 3y = 17$ is a tangent to a circle at the point $(2, 5)$.
The centre of the circle lies on the $y$-axis.
Find the coordinates of the centre of the circl... show full transcript
Worked Solution & Example Answer:The line $x + 3y = 17$ is a tangent to a circle at the point $(2, 5)$ - Scottish Highers Maths - Question 15 - 2023
Step 1
Determine gradient of tangent
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Answer
To find the gradient of the tangent line given by the equation x+3y=17, we can rearrange it into slope-intercept form (i.e., y=mx+b).
Starting from:
3y=−x+17
Dividing by 3 gives:
y=−31x+317
This shows that the gradient mt is −31.
Step 2
Determine gradient of radius
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Answer
The radius at the point (2,5) is perpendicular to the tangent line. The relationship between the gradients of two perpendicular lines is given by:
mt⋅mr=−1
where mt is the gradient of the tangent and mr is the gradient of the radius.
Substituting the known value:
−31imesmr=−1
Solving for mr gives:
mr=3.
Thus, the gradient of the radius is 3.
Step 3
Strategy to find centre
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Answer
Using the point-slope form of the equation of a line, where point (x0,y0)=(2,5) and the gradient is mr=3, the equation of the radius can be formed as follows:
y - 5 = 3(x - 2)$$
Rearranging this gives:
y = 3x - 6 + 5
= 3x - 1.\
Since the centre lies on the $y$-axis, the coordinate for the centre can be set as $(0, y_c)$, leading us to substitute $x = 0$:
y_c = 3(0) - 1
= -1.\
Step 4
State coordinates of centre
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Answer
Thus, the coordinates of the centre of the circle are:
(0,−1).
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