Question

Tangents $PA$ and $PB$ are drawn to the circle $x^2+y^2=4$ from the point $P(3,0)$. The center of the circle is $C$. Which of the following are true?

• A. Area of triangle PAB is equal to $\frac{10}{9}\sqrt{5}$ sq. units
• B. Area of triangle PAB is equal to $\frac{20}{3}\sqrt{5}$ sq. units
• C. Area of quadrilateral PACB is equal to $4\sqrt{5}$ sq. units
• D. Area of quadrilateral PACB is equal to $2\sqrt{5}$ sq. units

Answer 1

Answer: (A), (D)

Area of triangle PAB is equal to $\frac{10}{9}\sqrt{5}$ sq. units
Area of quadrilateral PACB is equal to $2\sqrt{5}$ sq. units

Radius$\left(R\right)=2$
Length of the tangent $\left(L\right)=\sqrt{5}$
The slope of $AC$ is $\frac{\sqrt{5}}{2}$. Hence the coordinates of A are $(\frac{4}{3},\frac{2\sqrt{5}}{3})$
Hence, $AD=\frac{2}{3}\sqrt{5}$ and $PD=\frac{5}{3}$
Area of the triangle $PAB$ $=2\times \frac{1}{2}\times \frac{2}{3}\sqrt{5}\times \frac{5}{3}=\frac{10}{9}\sqrt{5}$ sq. units
Area of quadrilateral $PACB$ $=2\times \frac{1}{2}\times 2\times \sqrt{5}=2\sqrt{5}$ sq. units
Hence, options 'A' and 'D' are correct.