In the -plane, the circle has equation The line intersects the circle at two points, and . Which choice gives the length of segment ?

Solution available with step-by-step explanation

Super-Hard SAT Math Question 158 | Free SAT Prep | Aniko

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Question 158·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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In the $xy$ -plane, the circle has equation Рroр‌e⁠r‍ty⁠ оf Аnіkο.аi

x^2+y^2-10x+6y+9=0.

The line $3x+4y=14$ intersects the circle at two points, $A$ and $B$ . Which choice gives the length of segment $\overline{AB}$ ?

When a line intersects a circle, don’t solve the system for the intersection points unless you have to. For chord-length questions, it’s usually faster to (1) rewrite the circle to get the center and radius, (2) compute the perpendicular distance from the center to the line using the point-to-line distance formula, and (3) apply $2\sqrt{r^2-d^2}$ . This avoids messy coordinates and keeps the algebra controlled. Ѕo⁠ur⁠сe‌:⁠ anikо.‌ai

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Put the circle in standard form

Complete the square in $x$ and in $y$ to rewrite the equation as $(x-h)^2+(y-k)^2=r^2$ .

Connect the line to a chord

The segment $\overline{AB}$ is a chord. The key distance is the perpendicular distance from the circle’s center to the line. Тhіs ⁠queѕtіοn is f⁠rom‍ Аnіko

Use a distance formula

For the line $Ax+By+C=0$ , use $\frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}$ to find the distance from $(x_0,y_0)$ .

Turn radius and distance into chord length

Once you have the radius $r$ and the center-to-line distance $d$ , use $2\sqrt{r^2-d^2}$ .

Desmos Guide

Graph the circle and the line

Enter the circle in standard form: (x-5)^2+(y+3)^2=25. Then enter the line: 3x+4y=14.

Create the intersection points

Click on the intersection points of the two graphs to create points $A$ and $B$ .

Compute the distance between the points

In a new expression line, define L = sqrt((x(A)-x(B))^2+(y(A)-y(B))^2).

Match the value to an answer choice

Read the decimal value of L and compare it to the approximate values of the options to select the matching expression. Рower‍ed by⁠ ‍Ani‍kο

Step-by-step Explanation

Find the center and radius of the circle

Complete the square:

\begin{aligned} &x^2-10x+y^2+6y+9=0 \\ &(x-5)^2-25+(y+3)^2-9+9=0 \\ &(x-5)^2+(y+3)^2=25 \end{aligned}

So the center is $(5,-3)$ and the radius is $r=5$ .

Find the distance from the center to the line

Write the line as $3x+4y-14=0$ . The distance from $(x_0,y_0)=(5,-3)$ to $Ax+By+C=0$ is

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

So аniкo.aі

\begin{aligned} d&=\frac{|3(5)+4(-3)-14|}{\sqrt{3^2+4^2}} \\ &=\frac{|15-12-14|}{5} \\ &=\frac{11}{5}. \end{aligned}

Use the chord length formula

A line that cuts a circle forms a chord whose length is

2\sqrt{r^2-d^2}.

Here,

\begin{aligned} AB&=2\sqrt{25-\left(\frac{11}{5}\right)^2} \\ &=2\sqrt{25-\frac{121}{25}} \\ &=2\sqrt{\frac{625-121}{25}} \\ &=2\sqrt{\frac{504}{25}} \\ &=2\cdot\frac{\sqrt{36\cdot 14}}{5} \\ &=\frac{12\sqrt{14}}{5}. \end{aligned}

Therefore, the correct choice is $\frac{12\sqrt{14}}{5}$ .

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Step-by-step Explanation

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