# Tangents are drawn to the hyperbola 4x2−y2=36 at the points P and Q. If these tangents intersect at the point T(0, 3) then the area (in sq. units) of ΔPTQ is : Option 1) 36$\sqrt{5}$ Option 2) 45${\sqrt{5}}$ Option 3) 54${\sqrt3}$ Option 4) 60${\sqrt3}$

H Himanshu

$hyperbola : \frac{x^{2}}{9}-\frac{y^{2}}{36}=1$

$T = 0\Rightarrow 0-\frac{3y}{36}=1$

$y=-12$

if $y=-12$  in $\frac{x^{2}}{9}-\frac{y^{2}}{36}= 1$

$x = \pm 3\sqrt5$

Thus $\Delta PTQ = 1/2 \times 6\sqrt5\times 15$

$=45\sqrt5$

Equation of Tangent to Hyperbola -

$\frac{xx_{1}}{a^{2}}-\frac{yy_{1}}{b^{2}}= 1$

- wherein

For the Hyperbola

$\frac{x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}= 1$ and

$P\left ( x_{1} ,y_{1}\right )$

Option 1)

36$\sqrt{5}$

This is incorrect

Option 2)

45${\sqrt{5}}$

This is correct

Option 3)

54${\sqrt3}$

This is incorrect

Option 4)

60${\sqrt3}$

This is incorrect

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