Question

The values of $\lambda$ for which the curve ${(7x+5)}^2+{(7y+3)}^2={\lambda }^2{(4x+3y-24)}^2$ represents a parabola is

• A. $\pm \frac{6}{5}$
• B. $\pm \frac{7}{5}$
• C. $\pm \frac{1}{5}$
• D. $\pm \frac{2}{5}$

Answer 1

The correct option is B $\pm \frac{7}{5}$

${(7x+5)}^2+{(7y+3)}^2={\lambda }^2{(4x+3y-24)}^2$

$\therefore {(x+\frac{5}{7})}^2+{(y+\frac{3}{7})}^2=(\frac{\lambda }{7}{)}^2{(4x+3y-24)}^2$

$\therefore {(x+\frac{5}{7})}^2+{(y+\frac{3}{7})}^2=(\frac{5\lambda }{7}{)}^2{\left(\frac{4x+3y-24}{5}\right)}^2$

$\therefore$ Distance of point from focus = e $\times$ Distance of the point from the directrix.

$⟹{\left(\frac{5\lambda }{7}\right)}^2=e=1$

$⟹\lambda =\pm \frac{7}{5}$

So, the answer is option (B).