Question

The two lines through (2, 3) from which the circle $x^2+y^2=25$ intercepts chords of length 8 units have equations

• A. $2x+3y=13,x+5y=17$
• B. $y=3,12x+5y=39$
• C. $x=2,9x-11y=51$
• D. None of these

Answer 1

The correct option is B $y=3,12x+5y=39$

The length of the chord intercepted by the circle $x^2+y^2=a^2$ on the straight line

$y=mx+c$ is $\sqrt[2]{2\frac{a^2(1+m^2)-c^2}{1+m^2}}$

Here $a=5$ and $y=mx+c$ passes through $(2,3)$, therefore $3=2m+c$ i.e. $c=3-2m.$

By hypothesis, $\sqrt[2]{2\frac{a^2(1+m^2)-c^2}{1+m^2}}=8$

$\Rightarrow \sqrt[2]{2\frac{25(1+m^2)-(3-2m{)}^2}{1+m^2}}=8$

(on putting the values of $a$ and $c$)

$\Rightarrow 5m^2+12m=0\Rightarrow m=0,-\frac{12}{5}$

$\Rightarrow c=3$ or $\frac{39}{5}$ (using $c=3-2m$).

Thus, the equations of the required lines are $y=3$ and $12x+5y=39$.