Question

Assertion :STATEMENT- 1 : A straight line passing through the origin intersects the branches of the hyperbola xy = 9 at Q in the first and R in the third quadrant. The point P divides the segment QR internally in the ratio 1:2, then the locus of P is ellipse Reason: STATEMENT- 2 : $A\equiv (x_1,y_1),B\equiv (x_2,y_2)$.

If P divides line joining Aand B in the ratio m : n, then co-ordinates of P are

$[\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n}]$

• A. Statement -1 is True, Statement -2 is True ; Statement -2 is a correct explanation for Statement -1
• B. Statement-1 is True, Statement-2 is True ; Statement-2 is NOT a correct explanation for Statement-1
• C. Statement -1 is True, Statement -2 is False
• D. Statement -1 is False, Statement -2 is True

Answer 1

The correct option is D Statement -1 is False, Statement -2 is True

STATEMENT-1 Let the line be y = mx and for R & Q

xy = 9

$\Rightarrow m{x}^2=9$

$\Rightarrow x=\frac{\pm 3}{\sqrt{m}}$ ; (m> 0)

for P(h, k)

$h=\frac{\frac{6}{\sqrt{m}}-\frac{3}{\sqrt{m}}}{3}=\frac{1}{\sqrt{m}}$

$k=\frac{3\sqrt{m}}{3}=\sqrt{m}$

$\Rightarrow$ hk =1 $\Rightarrow$ xy =1 is a hyperbola

Hence STATEMENT-1 is false.

STATEMENT- 2 : It is a standard formula.