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# Positional relationship between a circle and a line

Question:

A line 3x + 5y + 15 = 0 and a circle x2 + y2 - 8x - 6y - 11 = 0 lie in the xy-plane. Which of the following statement is true?

1. The line intersects the circle.

2. The line is tangent to the circle.

3. The line and the circle do not have intersection point.

Solution:

Assumed that the line has the equation, Ax + By + C = 0. The equation is a general formula of a line equation. P is a point, its x-coordinate is xo and y-coordinate is yo. Now we draw a line from the point P perpendicular to the line L at point Q. The distance from the point P to the point Q is |PQ| = d.

We use the formula, the distance from the point P(xo, yo) to a line, Ax + By + C = 0 is, d = |Axo + Byo + C|/square root of (A2 + B2)

If the point P is the center of the circle and r is the radius of the circle, then when d < r, the line intersects the circle, when d = r, then the line is tangent to the circle. When d > r, the line and the circle do not have intersection point.

Compare the given line equation 3x + 5y + 15 = 0 with the general line equation Ax + By + C = 0, we get, A = 3, B = 5 and C = 15.

Now we will find xo and yo which is the center of the circle.

The given circle equation is: x2 + y2 - 8x - 6y - 11 = 0, now we change the given circle equation,

x2 - 8x + 42 - 42 + y2 - 6y + 32 - 32 - 11 = 0
(x - 4)2 + (y - 3)2 = 42 + 32 + 11
(x - 4)2 + (y - 3)2 = 16 + 9 + 11
(x - 4)2 + (y - 3)2 = 36
(x - 4)2 + (y - 3)2 = 62

Therefore, the center of the circle is (4, 3) and the radius of the circle is 6. So, xo = 4, yo = 3, and r = 6.

Now, we will find the distance from the center of the circle to the line, we use the formula,

d = |Axo + Byo + C|/square root of (A2 + B2)
= |3 × 4 + 5 × 3 + 15|/square of 32 + 52
= |12 + 15 + 15|/square root of 9 + 25
= 42/square root of 34
= 42/5.83
= 7.2

Because d > r, so, the line is not intersecting the circle. The statement 3 is correct. Watch the video for more details.

Note: If the distance from the center of the circle to the line L is less than the radius of the circle, then the line intersects the circle. If the distance from the center of the circle to the line L is equal to the radius of the circle, then the line is tangent to the circle.