Question

What is the equation of the chord of the circle $x^2+y^2=81$ which is bisected at point $\left(-2,3\right)$ ?

• A. $3x-y=13$
• B. $3x-4y=13$
• C. $2x-3y=13$
• D. $3x-3y=13$
• E. $2x-3y=-13$

Answer 1

The correct option is E

$2x-3y=-13$

Explanation of correct answer :

Finding equation of chord of circle.

Given, equation of circle is $x^2+y^2=81$.

As we know, if the equation of circle is $x^2+y^2=r^2$

Then, Radius = $r$, Centre = $\left(0,0\right)$.

Let the equation of chord is $y=mx+c\dots \left(1\right)$

The line joining the midpoint of the chord to the centre of the circle is perpendicular to the circle, who slope is,

$$\begin{gathered} m=\frac{x_2-x_1}{y_2-y_1} \\ \Rightarrow m=\frac{2-0}{3-0} \\ \Rightarrow m=\frac{2}{3} \end{gathered}$$

To find the value of $c$, put the value $\left(-2,3\right)$ in the equation $\left(1\right)$

$$\begin{gathered} \Rightarrow 3=\frac{2}{3}\times \left(-2\right)+c \\ \Rightarrow c=\frac{13}{3} \end{gathered}$$

Put the values of $m,c$ in equation $\left(1\right)$ to get the equation of chord.

$$\begin{gathered} \Rightarrow y=\frac{2}{3}x+\frac{13}{3} \\ \Rightarrow 2x-3y+13=0 \end{gathered}$$

Hence, the equation of chord is $2x-3y=-13$

Thus, the correct answer is option (E).