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S_A_V [24]
2 years ago
15

Find the equation of the circle having radius 5 and which touches the line 3x+4y-16=0 at point (4,1)?

Mathematics
1 answer:
Sladkaya [172]2 years ago
5 0

The only information missing is the circle's center; let it be the point (a, b). Then the equation of the circle is

(x - a)² + (y - b)² = 5²

We know the point (4, 1) lies on the circle, so we have the condition

(4 - a)² + (1 - b)² = 5²   ⇒   (a - 4)² + (b - 1)² = 25

The circle just touches the line, so the line is tangent to the circle. The slope of the line is -3/4, since

3x + 4y - 16 = 0   ⇒   y = 4 - 3/4 x

Any non-vertical tangent line to the circle at (a', b') has slope equal to dy/dx with x = a' and y = b'. Differentiating both sides of the circle equation with respect to x yields

2 (x - a) + 2 (y - b) dy/dx = 0   ⇒   dy/dx = -(x - a)/(y - b)

At the point (4, 1), the slope of the tangent is -3/4, so we get the condition

-(4 - a)/(1 - b) = -3/4   ⇒   4a - 3b = 13

Solve for b in terms of a and substitute into the other condition to solve for a :

4a - 3b = 13   ⇒   b = (4a - 13)/3

(a - 4)² + ((4a - 13)/3 - 1)² = 25

25/9 a² - 200/9 a + 400/9 = 25

25/9 a² - 200/9 a + 175/9 = 0

a² - 8a + 7 = 0

(a - 7) (a - 1) = 0

a = 7   or   a = 1

Solve for b :

b = (4a - 13)/3   ⇒   b = 5   or   b = -3

So there are two possible circles,

(x - 7)² + (y - 5)² = 5²

or

(x - 1)² + (y + 3)² = 5²

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