Question 928·Hard·Math

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In the $xy$ -plane, the circle with equation $x^2 + y^2 = 13$ and the line with equation $3y + 2x = k$ , where $k$ is a positive constant, intersect at exactly one point. What is the value of $k$ ?

Your answer

A line and circle with exactly one intersection are tangent: the line touches the circle without crossing it. Read the center and radius, then use the line’s coefficients to find its perpendicular distance from the center. Desmos can solve the resulting equation by regression. At tangency, that distance equals the radius. Passing through a point on the circle alone isn’t enough.

Hints

Hint 1
A tangent line touches a circle once. A line closer to the center cuts through it twice. For one touch, how far from the center must the line be?
Hint 2
In $x^2+y^2=r^2$ , the center is $(0,0)$ and the right side is the radius squared, not the radius. What radius does the given equation give you?
Hint 3
For $Ax+By=C$ , the perpendicular distance from $(0,0)$ to the line is $\frac{|C|}{\sqrt{A^2+B^2}}$ . Use the condition that $k$ is positive, then set this distance equal to the radius.

Step-by-step

Match the line’s distance to the radius
Step 1
See what one intersection requires
Set $k=1$ as a sample positive value, then type both equations as written. Desmos draws the circle and the line, with a slider for $k$ . At $k=1$ , the line crosses the circle twice. For exactly one intersection, it must instead touch the circle without cutting through it.
Step 2
Find the circle’s radius
The form $x^2+y^2=r^2$ describes a circle centered at $(0,0)$ , where $r$ is its radius. Match the right side to $r^2$ , then take the positive square root:
$r^2=13$ $r=\sqrt{13}.$
Don’t use $13$ as the radius; it’s the radius squared.
Step 3
Find the line’s distance from the center
Rewrite the line as $2x+3y=k$ . Its coefficient pair $(2,3)$ points perpendicular to it and has length $\sqrt{2^2+3^2}$ . Type that length into Desmos; it shows about $3.606$ . The perpendicular distance $d$ from $(0,0)$ to the line is the gap in its equation divided by that length:
$d=\frac{|2(0)+3(0)-k|}{\sqrt{2^2+3^2}}.$
Evaluate the zeros and the length:
$d=\frac{|-k|}{\sqrt{13}}.$
Use $k>0$ to remove the absolute value:
$d=\frac{k}{\sqrt{13}}.$
Step 4
Turn one touch into an equation
A line closer than the radius crosses the circle twice; farther away, it misses. So one touch means its shortest distance from the center equals the radius:
$\frac{k}{\sqrt{13}}=\sqrt{13}.$
Passing through some point on the circle would not, by itself, rule out a second intersection.
Step 5
Solve for the positive constant
Type $\frac{k_1}{\sqrt{13}}\sim\sqrt{13}$ into Desmos. The subscript makes $k_1$ a value for Desmos to find, and $\sim$ asks it to fit the two sides. Under PARAMETERS, Desmos reports $k_1=13$ . So the positive value of $k$ that makes the line touch the circle once is $13$ . Grid in 13.

Question 928·Hard·Math

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