In the xy-plane, a circle has center (6, 8) and is tangent to the y-axis. A line with a po… — SAT

In the $xy$ -plane, a circle has center $(6, 8)$ and is tangent to the $y$ -axis. A line with a positive slope $m$ passes through the origin and is tangent to the circle. If $m$ is written as a fraction in simplest form, $\frac{p}{q}$ , what is the value of $p + q$ ?

Answer: 31

Answer

The value of

p + q

is 31.

The radius of the circle is 6 since the center is

(6, 8)

and it is tangent to the

y

-axis. Substituting the line

y = mx

into the circle's equation

(x-6)^2 + (y-8)^2 = 36

yields a quadratic equation in

x

(1+m^2)x^2 - (12+16m)x + 64 = 0

. Since the line is tangent to the circle, there is exactly one solution, meaning the discriminant of this quadratic must equal zero:

(12+16m)^2 - 4(1+m^2)(64) = 0

. Simplifying this equation yields

24m - 7 = 0

, which gives a slope of

m = \frac{7}{24}

. Since the fraction is in simplest form,

p=7

and

q=24

, and their sum is

31

Step-by-Step Solution

Determine the radius of the circle.

r = 6

Since the center is at

(6, 8)

and the circle is tangent to the

y

-axis, the radius is the horizontal distance from the center to the

y

-axis.

Write the equation of the circle.

(x-6)^2 + (y-8)^2 = 36

A circle with center

(h, k)

and radius

r

has the equation

(x-h)^2 + (y-k)^2 = r^2

Substitute

y = mx

into the circle's equation.

(1+m^2)x^2 - (12+16m)x + 64 = 0

Substituting the equation of the line

y = mx

into the circle's equation and expanding gives a quadratic equation in terms of

x

Set the discriminant of the quadratic equation to zero.

\Delta = (12+16m)^2 - 256(1+m^2) = 0

For the line to be tangent to the circle, the system must have exactly one real solution, meaning the quadratic equation must have a discriminant of zero.

Solve for the positive slope

m

m = \frac{7}{24}

Expanding and simplifying the discriminant equation yields

24m - 7 = 0

, which gives

m = \frac{7}{24}

Calculate

p + q

using the simplest form fraction.

31

Since

m = \frac{7}{24}

is in simplest form,

p = 7

and

q = 24

. Therefore,

p + q = 7 + 24 = 31

Key Concept

Nonlinear Systems of Equations

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic