Question

The measure of the chord intercepted circle $x^2+y^2=9$ and the line $x-y+2=0$ is

• A. $\sqrt{28}$
• B. $2\sqrt{5}$
• C. $7$
• D. $5$

Answer 1

The correct option is A

$\sqrt{28}$

Explanation for the correct option:

Step 1: Find the points of intersection.

A circle $x^2+y^2=9$ and a line $x-y+2=0$ is given.

So, we can rewrite the equation of line as follows $x=y-2...1$.

Now, from the equation of the circle, we get

$$\begin{gathered} {(y-2)}^2+y^2=9 \\ \Rightarrow y^2+4-4y+y^2=9 \\ \Rightarrow 2y^2-4y-5=0 \end{gathered}$$

Use quadratic fornula.

$$\begin{gathered} y=\frac{4\pm \sqrt{4^2-4\left(2\right)(-5)}}{2\left(2\right)} \\ \Rightarrow y=\frac{4\pm \sqrt{16+40}}{4} \\ \Rightarrow y=\frac{4\pm \sqrt{56}}{4} \\ \Rightarrow y=\frac{4\pm 2\sqrt{14}}{4} \\ \Rightarrow y=\frac{2\pm \sqrt{14}}{2} \end{gathered}$$

Therefore, the $y-$coordinates of the points of intersection are $\frac{2-\sqrt{14}}{2},\frac{2+\sqrt{14}}{2}$.

From equation $1$.

When $y=\frac{2-\sqrt{14}}{2}$.

$$\begin{gathered} x=\frac{2-\sqrt{14}}{2}-2 \\ \Rightarrow x=\frac{2-\sqrt{14}-4}{2} \\ \Rightarrow x=\frac{-2-\sqrt{14}}{2} \end{gathered}$$

Therefore, when $y=\frac{2-\sqrt{14}}{2}$, $x=\frac{-2-\sqrt{14}}{2}$.

When $y=\frac{2+\sqrt{14}}{2}$.

$$\begin{gathered} x=\frac{2+\sqrt{14}}{2}-2 \\ \Rightarrow x=\frac{2+\sqrt{14}-4}{2} \\ \Rightarrow x=\frac{-2+\sqrt{14}}{2} \end{gathered}$$

Therefore, when $y=\frac{2+\sqrt{14}}{2}$, $x=\frac{-2+\sqrt{14}}{2}$.

Therefore, the point of intersections are $\left(\frac{-2-\sqrt{14}}{2},\frac{2-\sqrt{14}}{2}\right)$ and $\left(\frac{-2+\sqrt{14}}{2},\frac{2+\sqrt{14}}{2}\right)$.

Step 2: Compute the length of the chord.

We know that the length of the chord $l$ is equal to the distance between the points of intersection of the circle and the line.

So,

$$\begin{gathered} l=\sqrt{{\left(\frac{-2-\sqrt{14}}{2}-\frac{-2+\sqrt{14}}{2}\right)}^2+{\left(\frac{2-\sqrt{14}}{2}-\frac{2+\sqrt{14}}{2}\right)}^2} \\ \Rightarrow l=\sqrt{{\left(\frac{-2\sqrt{14}}{2}\right)}^2+{\left(\frac{-2\sqrt{14}}{2}\right)}^2} \\ \Rightarrow l=\sqrt{14+14} \\ \Rightarrow l=\sqrt{28} \end{gathered}$$

Therefore, the length of the chord is $\sqrt{28}$.

Hence, option $A$ is the correct answer.