Question

Assertion :If points $A(c{t}_1,\frac{c}{t_1})$, $B(c{t}_2,\frac{c}{t_2})$, $C(c{t}_3,\frac{c}{t_3})$ and $D(c{t}_4,\frac{c}{t_4})$ lies on the circle $x^2+y^2+2gx+2fy+k=0$ then $t_1+t_2+t_3+t_4=-\frac{2g}{c}$. Reason: Points $A,B,C,D$ are parametric points of $xy=c^2$.

• 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

Reason (R) is true as $A\left(c{t}_1\frac{c}{t_1}\right),B(c{t}_2,\frac{c}{t_2})C(c{t}_3,\frac{c}{t_3})$ & $D(c{t}_4,\frac{c}{t_4})$
are parametric points and satisfies $xy=c^{2}i.e.xy=c{t}_1\times \frac{c}{t_1}=c^2$
Suppose the circle $x^2+y^2+2gx+2fy+k=0$ cuts the hyperbola at $(ct,\frac{c}{t})$
$\therefore c^2{t}^2+\frac{c^2}{t^2}+2gct+\frac{2fc}{t}+k=0$
$\Rightarrow c^2{t}^4+2gc{t}^3+k{t}^2+2kct+c^2=0$
Which has four roots say $t_1,t_2,t_3,t_4$
$\therefore t_1+t_2+t_3+t_4=-\frac{2gc}{c^2}=-\frac{2g}{c}$
Hence assertion (A) is true follows the reason (R),