A circle has equation , where is a constant. The line is tangent to the circle. Which choice gives the value of ?

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Super-Hard SAT Math Question 88 | Free SAT Prep | Aniko

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Question 88·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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A circle has equation $x^2 + y^2 - 6x + 8y = m$ , where $m$ is a constant. The line $3x + 4y = 24$ is tangent to the circle.

Which choice gives the value of $m$ ? ©‌ аn⁠iк‌о.аi

For circle-and-line tangency in coordinate geometry, first rewrite the circle in standard form to get the center and radius. Then use the key tangency fact: the radius equals the perpendicular distance from the center to the line. Setting those equal usually creates a clean equation for the unknown parameter. Аn‍іko Queѕtіo⁠n Bаnк

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Find the center and radius

Rewrite $x^2 + y^2 - 6x + 8y = m$ in the form $(x-h)^2+(y-k)^2=r^2$ by completing the square for $x$ and for $y$ . From anік⁠o.аі

Use what “tangent” means

If a line is tangent to a circle, the perpendicular distance from the circle’s center to the line equals the circle’s radius.

Distance from a point to a line

For a line $Ax+By=C$ , the distance from $(x_0,y_0)$ to the line is

\frac{|Ax_0+By_0-C|}{\sqrt{A^2+B^2}}.

Desmos Guide

Graph the line

Enter the line as 3x+4y=24.

Create a slider for m and graph the circle

Type m=1 to create a slider.

Then enter the circle equation as x^2+y^2-6x+8y=m.

Adjust m until the line is tangent

Move the slider for $m$ until the line touches the circle at exactly one point (zoom in near the touching point to check that it is not crossing).

Match the slider value to an answer choice

Read the value of $m$ from the slider (it should be about $13.44$ ) and choose the option that equals that value. From аnіko.аi

Step-by-step Explanation

Put the circle in standard form

Complete the square for $x$ and for $y$ : Тh‍iѕ queѕtiоn is frоm Аn‍іко

\begin{aligned} x^2-6x + y^2+8y &= m \\ (x-3)^2-9 + (y+4)^2-16 &= m \\ (x-3)^2 + (y+4)^2 &= m+25 \end{aligned}

So the center is $(3,-4)$ and the radius is $r=\sqrt{m+25}$ .

Find the distance from the center to the line

A line tangent to a circle is exactly one radius away from the circle’s center.

For the line $3x+4y=24$ , the distance from $(3,-4)$ to the line is

\frac{|3(3)+4(-4)-24|}{\sqrt{3^2+4^2}} = \frac{|9-16-24|}{5} = \frac{31}{5}.

Set radius equal to that distance and solve

Tangency gives

\sqrt{m+25}=\frac{31}{5}.

Square both sides:

\begin{aligned} m+25 &= \left(\frac{31}{5}\right)^2=\frac{961}{25} \\ m &= \frac{961}{25}-\frac{625}{25}=\frac{336}{25}. \end{aligned}

So the value of $m$ is $\frac{336}{25}$ .

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