The line $y = 3x + 7$ intersects the circle $x^2 + y^2 - 4x - 6y - 7 = 0$ at the points P and Q. (a) Find the coordinates of P and Q. PQ is a tangent to a second, smaller circle. This circle is concentric with the first. (b) Determine the equation of the smaller circle.

Scottish Highers Maths Circle: The line $y = 3x + 7$ intersects

The line $y = 3x + 7$ intersects the circle $x^2 + y^2 - 4x - 6y - 7 = 0$ at the points P and Q - Scottish Highers Maths - Question 9 - 2022

Question 9

The line $y = 3x + 7$ intersects the circle $x^2 + y^2 - 4x - 6y - 7 = 0$ at the points P and Q. (a) Find the coordinates of P and Q. PQ is a tangent to a second, ... show full transcript

Worked Solution & Example Answer:The line $y = 3x + 7$ intersects the circle $x^2 + y^2 - 4x - 6y - 7 = 0$ at the points P and Q - Scottish Highers Maths - Question 9 - 2022

Step 1

Find the coordinates of P and Q

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Answer

To find the coordinates of points P and Q where the line intersects the circle, we first substitute the equation of the line into the circle's equation:

Substitute $y = 3x + 7$ into the circle's equation: $x^2 + (3x + 7)^2 - 4x - 6(3x + 7) - 7 = 0$
Expand the equation: [ x^2 + (9x^2 + 42x + 49) - 4x - (18x + 42) - 7 = 0 ] Simplifying this gives: [ 10x^2 + 20x = 0 ]
Factor the equation: $10x(x + 2) = 0$ Therefore, $x = 0$ or $x = -2$ .
Calculate the corresponding $y$ coordinates:
- For $x = 0$ : $y = 3(0) + 7 = 7$ So point P is $(0, 7)$ .
- For $x = -2$ : $y = 3(-2) + 7 = 1$ So point Q is $(-2, 1)$ .

Step 2

Determine the equation of the smaller circle

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Answer

Identify the center of the circle from part (a), which is at $(2, 3)$ .
Calculate the midpoint of PQ: $\text{Midpoint} = \left( \frac{0 + (-2)}{2}, \frac{7 + 1}{2} \right) = (-1, 4)$
Calculate the radius of the smaller circle, which is the distance from the center $(2, 3)$ to the tangent point Q $(-2, 1)$ : $r = \sqrt{(2 - (-2))^2 + (3 - 1)^2} = \sqrt{(4)^2 + (2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$
The equation of the smaller circle is: $(x - 2)^2 + (y - 3)^2 = (2\sqrt{5})^2 = 20$ Thus, the final equation is: $(x - 2)^2 + (y - 3)^2 = 20$ .

The line $y = 3x + 7$ intersects the circle $x^2 + y^2 - 4x - 6y - 7 = 0$ at the points P and Q - Scottish Highers Maths - Question 9 - 2022

Worked Solution & Example Answer:The line $y = 3x + 7$ intersects the circle $x^2 + y^2 - 4x - 6y - 7 = 0$ at the points P and Q - Scottish Highers Maths - Question 9 - 2022

Find the coordinates of P and Q

Determine the equation of the smaller circle

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