Question

Assertion :The center of the mean position of the four points bisects the distance between the centre of the two curves. Reason: $\frac{x_1+x_2+x_3+x_4}{4}=\frac{g}{2},\frac{y_1+y_2+y_3+y_4}{4}=\frac{f}{2},x_1.x_2.x_3.x_4=y_1.y_2.y_3.y_4=1$.

• 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 A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Let the rectangular hyperbola be $xy=c^2$ and the equation of the circle be $x^2+y^2+2gx+2fy+k=0$.

Any point on the hyperbola is $(cp,\frac{c}{p})$. If it lies on the circle, then $c^2{p}^2+\frac{c^2}{p2}+2gcp+2f\frac{c}{p}+k=0$. $c^2{p}^4+2gc{p}^3+k{p}^2+2fcp+c^2=0$.

This is fourth-degree equation in $p$, which has four roots. Hence the circle and the hyperbola intersect in four points.

If $p_1,p_2,p_3,p_4$ are the roots of this equation, then$p_1+p_2+p_3+p_4=\frac{2gc}{c^2}=\frac{2g}{c}$ $c{p}_1+c{p}_2+c{p}_3+c{p}_4=2g$ $\frac{x_1+x_2+x_3+x_4}{4}=\frac{g}{2}$

Also $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}+\frac{1}{p_4}=\frac{p_1.p_2.p_3}{p_1.p_2.p_3.p_4}=\frac{2fc}{c^2}=\frac{2f}{c}$ $\frac{c}{p_1}+\frac{c}{p_2}+\frac{c}{p_3}+\frac{c}{p_4}=2f$ $\frac{y_1+y_2+y_3+y_4}{4}=\frac{f}{2}$.

Hence the mean of the four points is $(\frac{g}{2},\frac{f}{2})$ which is the mid-point of the center of the hyperbola and that of the circle. Hence, the correct option is (a).