Question

Assertion :If the equation $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represent two straight lines, then product of perpendiculars drawn from the origin to these lines is $\left|\frac{c}{\sqrt{{(a-b)}^2+4h^2}}\right|$ Reason: The perpendicular distance from origin to line $Ax+By+k=0$ is $\frac{\left|k\right|}{\sqrt{A^2+B^2}}$

• 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 B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

We have $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=\mathrm{0....}\left(i\right)$ Let $lx+my+n=0....\left(ii\right)$ $l^{\prime }x+m^{\prime }y+n^{\prime }=0.....\left(iii\right)$ Then $(lx+my+n)(l^{\prime }x+m^{\prime }y+n^{\prime })=a{x}^2+2hxy+b{y}^2+2gx+2fy+c$ On comparing the coefficients of like terms, we .have $l{l}^{\prime }=a,m{m}^{\prime }=b,n{n}^{\prime }=c$ and $l{m}^{\prime }+l^{\prime }m=2h,l{n}^{\prime }+l^{\prime }n=2g,m{n}^{\prime }+m^{\prime }n=2f$ Let $P_1,P_2$ be the lengths of. $\perp$ er drawn from (0,0) on the lines (ii) & (iii) respectively, then $P_1=\frac{|0+0+n|}{\sqrt{l^2+m^2}},P_2=\frac{\left|n^{\prime }\right|}{\sqrt{l^{\prime 2}+m^{\prime 2}}}$
$\therefore P_1{P}_2=\left|\frac{n{n}^{\prime }}{\sqrt{(l^2+m^2)(l^{\prime 2}+m^{\prime 2})}}\right|$
$=\left|\frac{n{n}^{\prime }}{\sqrt{(l{l}^{\prime }{)}^2+(m{m}^{\prime }{)}^2+(l{m}^{\prime }{)}^2+(l^{\prime }m{)}^2}}\right|$ $=\left|\frac{n{n}^{\prime }}{\sqrt{(l{l}^{\prime }-m{m}^{\prime }{)}^2+(l^{\prime }m+m^{\prime }l{)}^2}}\right|$ $=\frac{\left|c\right|}{\sqrt{(a-b{)}^2+4h^2}}$