In the standard (x, y) coordinate plane, a circle is defined by the equation x^2 + y^2 - 1… — ACT

In the standard $(x, y)$ coordinate plane, a circle is defined by the equation $x^2 + y^2 - 10x - 4y = 0$ . The line $y = x$ intersects the circle at the origin $(0, 0)$ and at a second point $P$ . What is the $x$ -coordinate of point $P$ ?

Answer: 7

Answer

The correct answer is 7.

To find the points of intersection between the circle and the line, we substitute the line's equation

y = x

into the circle's equation

x^2 + y^2 - 10x - 4y = 0

. This substitution yields

2x^2 - 14x = 0

. Factoring out

2x

gives

2x(x - 7) = 0

, which has solutions

x = 0

and

x = 7

. The solution

x = 0

corresponds to the origin

(0, 0)

. Therefore, the

x

-coordinate of the second point

P

7

Step-by-Step Solution

Substitute

y = x

into the equation of the circle.

x^2 + x^2 - 10x - 4x = 0

Since the line is

y = x

, any point of intersection must satisfy both equations.

Combine like terms to simplify the quadratic equation.

2x^2 - 14x = 0

Grouping the quadratic and linear terms yields a single equation in terms of

x

Factor the quadratic equation.

2x(x - 7) = 0

Factoring allows us to identify the individual roots of the equation.

Solve for

x

and identify the value for point

P

x = 7

The solution

x = 0

corresponds to the origin. Thus, the

x

-coordinate of the second point

P

must be

7

Key Concept

Finding points of intersection between a line and a circle in the coordinate plane.

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