The equation of a circle is x^2 + y^2 = 25, and the equation of a line is 2x - y = 5. If t… — ACT

The equation of a circle is $x^2 + y^2 = 25$ , and the equation of a line is $2x - y = 5$ . If the line intersects the circle at the points $P$ and $Q$ , what is the sum of the $y$ -coordinates of $P$ and $Q$ ?

A
-18
B
-6
-2Answer
D
2
E
4

Answer

The sum of the

y

-coordinates is

-2

Substituting the linear equation into the circle equation yields a quadratic equation with roots

x = 0

and

x = 4

. Evaluating the linear equation at these values gives the

y

-coordinates

-5

and

3

. Adding these

y

-coordinates results in

-2

Step-by-Step Solution

Rearrange the linear equation to express

y

in terms of

x

y = 2x - 5

Expressing one variable in terms of the other allows for substitution into the quadratic circle equation.

Substitute

y = 2x - 5

into the circle equation

x^2 + y^2 = 25

and simplify.

x^2 + (2x - 5)^2 = 25 \implies x^2 + 4x^2 - 20x + 25 = 25 \implies 5x^2 - 20x = 0

This substitution reduces the system of equations to a single quadratic equation in terms of

x

Solve the quadratic equation

5x^2 - 20x = 0

for

x

5x(x - 4) = 0 \implies x_1 = 0

and

x_2 = 4

Finding the roots of this quadratic equation gives the

x

-coordinates of the intersection points.

Substitute the

x

-values back into the linear equation

y = 2x - 5

to find the corresponding

y

-coordinates.

For

x_1 = 0

y_1 = 2(0) - 5 = -5

. For

x_2 = 4

y_2 = 2(4) - 5 = 3

. The intersection points are

P(0, -5)

and

Q(4, 3)

This step determines the coordinates of the two points of intersection.

Calculate the sum of the

y

-coordinates of the points

P

and

Q

y_1 + y_2 = -5 + 3 = -2

This yields the final value requested by the question.

Key Concept

Solving a system of linear and circular equations by substitution

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic