Question

a circle is tangent to lines 5x+2y-10=0 and 5x+2y+2=0. find its area and center.

Answer 1

Answer on Question #59281 – Math – Analytic Geometry

Question

a circle is tangent to lines $5x+2y-10=0$ and $5x+2y+2=0$. find its area and center.

Solution

Method 1

These straight lines are parallel, because their slopes are equal, so there can be the infinite number of circles.

Take a point $(x_A; y_A) = (2; 0)$ which lies on the straight line $5x+2y-10=0$.

The distance between straight lines $5x+2y-10=0$ and $5x+2y+2=0$ is equal to the distance between the point $(x_A; y_A) = (2; 0)$ and the straight line $5x+2y+2=0$ :

$$d = \frac{|5 \cdot 2 + 2 \cdot 0 + 2|}{\sqrt{5^2 + 2^2}} = \frac{12}{\sqrt{29}}.$$

The length of circle’s radius is equal to

$$r = \frac{d}{2} = \frac{12}{2\sqrt{29}} = \frac{6}{\sqrt{29}}.$$

The area of the circle is equal to

$$S = \pi r^2 = \pi \left(\frac{6}{\sqrt{29}}\right)^2 = \frac{36\pi}{29}$$

Method 2

A circle is tangent to lines $5x + 2y - 10 = 0$ and $5x + 2y + 2 = 0$, so the equations of the lines are: $y = -2.5x + 5$, $y = -2.5x - 1$.

These lines are parallel, because their slopes are equal, so there can be the infinite number of circles with the centers on the line $y = -2.5x + 2$.

As the slope equals -2.5, then tangent of this angle is $\tan(a) = -2.5$, where $a$ is an angle between the line and the $x$-axis, so the angle is $111.8{}^\circ$. Let’s consider the rectangular triangle between the lines

$y = -2.5x - 1$ and $y = -2.5x + 5$, one its cathetus is the diameter of the circle, the hypotenuse equals 6 (distance between two lines, which is parallel to the $y$-axis, for example, distance between points (0; -1) and (0; 5) which lie on the lines given), its angle between cathetus, which is the diameter of the circle, and hypotenuse equals to the smaller angle between the line $y = -2.5x + 5$ and $x$-axis, because our rectangular triangle is similar to another rectangular triangle between the line

$y = -2.5 + 5$, $x$-axis and $y$-axis (3 equal angles), so that angle equals $180{}^\circ - a$, then

$$\frac{2r}{6} = \cos(180{}^\circ - a)$$

and the length of the radius of the circle is

$$r = 6 \cos(180{}^\circ - a)/2 = 3 \cos(68.2{}^\circ) = 3 \cdot 0.371 \approx 1.113$$

The area of the circle is $S = \pi \cdot r^2 = \pi \cdot 1.113^2 \approx 3.89$.

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