A circle in the standard (x, y) coordinate plane is represented by the equation x^2 + y^2 … — ACT

A circle in the standard $(x, y)$ coordinate plane is represented by the equation $x^2 + y^2 + 4x - 6y - 12 = 0$ . What is the distance between the two points where this circle intersects the $x$ -axis?

Answer: 8 units

Answer

The distance between the two points where the circle intersects the

x

-axis is 8 units.

Setting

y = 0

gives the quadratic equation

x^2 + 4x - 12 = 0

. Factoring yields

(x + 6)(x - 2) = 0

, giving solutions

x = -6

and

x = 2

. The distance between these two

x

-intercepts is

2 - (-6) = 8

units.

Step-by-Step Solution

Substitute

y = 0

into the circle equation.

x^2 + 4x - 12 = 0

Points on the

x

-axis have a

y

-coordinate of 0.

Solve the quadratic equation for

x

by factoring.

x = -6

and

x = 2

Factoring

(x + 6)(x - 2) = 0

yields the

x

-coordinates of the intersection points.

Find the distance between

(-6, 0)

and

(2, 0)

2 - (-6) = 8

The horizontal distance between two points

(x_1, 0)

and

(x_2, 0)

|x_2 - x_1|

Key Concept

Finding

x

-intercepts of a circle given in general form

Question

Answer

Step-by-Step Solution

Key Concept

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