In the standard (x, y) coordinate plane, a circle is defined by the equation x^2 + y^2 = 2… — ACT

In the standard $(x, y)$ coordinate plane, a circle is defined by the equation $x^2 + y^2 = 25$ and a line is defined by the equation $3x + 4y = 15$ . The line intersects the circle at two points, $A$ and $B$ . What is the distance between point $A$ and point $B$ ?

Answer: 8

Answer

The distance between the two intersection points is 8.

The correct answer is 8. We can find the intersection points by substituting

y = \frac{15 - 3x}{4}

into the circle's equation

x^2 + y^2 = 25

, yielding the quadratic equation

5x^2 - 18x - 35 = 0

. Solving this gives

x = 5

and

x = -1.4

, with corresponding

y

-coordinates

y = 0

and

y = 4.8

. The distance between

(5, 0)

and

(-1.4, 4.8)

\sqrt{(-1.4 - 5)^2 + (4.8 - 0)^2} = \sqrt{6.4^2 + 4.8^2} = \sqrt{64} = 8

. Alternatively, we can use geometry: the distance from the center of the circle

(0,0)

to the line

3x + 4y - 15 = 0

d = \frac{|3(0) + 4(0) - 15|}{\sqrt{3^2 + 4^2}} = 3

. Since the radius of the circle is

r = 5

, the right triangle formed by the radius, the perpendicular segment, and half the chord has a half-chord length of

\sqrt{5^2 - 3^2} = 4

. Thus, the total chord length is

2 \times 4 = 8

Step-by-Step Solution

Express the linear equation in terms of one variable

y = \frac{15 - 3x}{4}

This allows for substitution into the equation of the circle.

Substitute the expression into the circle's equation and simplify

x^2 + \left(\frac{15 - 3x}{4}\right)^2 = 25 \Rightarrow 16x^2 + (225 - 90x + 9x^2) = 400 \Rightarrow 25x^2 - 90x - 175 = 0 \Rightarrow 5x^2 - 18x - 35 = 0

To create a single quadratic equation in terms of

x

representing the intersection points.

Solve the quadratic equation for

x

(5x + 7)(x - 5) = 0 \Rightarrow x = 5

x = -1.4

To find the

x

-coordinates of the intersection points.

Calculate the corresponding

y

-coordinates

For

x = 5

y = 0

, giving point

A(5, 0)

. For

x = -1.4

y = 4.8

, giving point

B(-1.4, 4.8)

To determine the exact coordinates of both intersection points.

Apply the distance formula to find the length of the segment

AB

d = \sqrt{(-1.4 - 5)^2 + (4.8 - 0)^2} = \sqrt{(-6.4)^2 + 4.8^2} = \sqrt{40.96 + 23.04} = \sqrt{64} = 8

To compute the final distance between the two intersection points.

Key Concept

Solving systems of linear and quadratic equations to determine intersection points and calculating the distance between coordinates.

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