Question

Assertion :Consider $L_1:2x+3y+p-3=0$ $L_2:2x+3y+p+3=0.$ where p is a real number and $C:x^2+y^2+6x-10y+30=0$

If the line $L_1$ is a chord of the circle C then the line is $L_2$ not always a diameter of the circle C and

Reason: If the line $L_1$ is a diameter of the circle C then the line $L_2$ is not a chord of the circle C.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

Equation of given circle C is
${(x-3)}^2+{(y+5)}^2=9+25-30$
$ie,{(x-3)}^2+{(y+5)}^2=2^2$
Centre $=(3,-5)$
If $L_1$ is diameter, them $2\left(3\right)+3(-5)+p-3=0$
$\Rightarrow p=12$
$\therefore L_1$ is $2x+3y+9=0$
$L_2$ is $2x+3y+15=0$
Distance of centre of circle from $L_2$ equals
$\left|\frac{2\left(3\right)+3(-5)+15}{\sqrt{2^2+3^2}}\right|=\frac{6}{\sqrt{13}}<2$ (radius of circle)
$\therefore L_2$ is a chord of circle C.
Statement II,false