From the point $P (4,3)$ tangents $PA$ and $PB$ are drawn to the circle ${ x }^{ 2 }+{ y }^{ 2 }=9$ Than area of the triangle $PAB$ is-
- $\dfrac{192}{25}$sq units
- $\dfrac {16}{9}$ sq units
- $\dfrac { 21\sqrt { 3 } }{ 16\quad } sq\quad units$
- $\dfrac { 25\sqrt { 3 } }{ 9\quad } sq\quad units$
For a circle x^2 + y^2 = r^2, the length of the tangent from (x1, y1) is sqrt(x1^2 + y1^2 - r^2). Here, length L = sqrt(4^2 + 3^2 - 3^2) = 4. The area of triangle PAB is (L^3 * r) / (L^2 + r^2) = (64 * 3) / (16 + 9) = 192 / 25.
The circle x^2 + y^2 = 9 has its center at the origin (0, 0) and a radius of r = 3. The distance d from the point P(4, 3) to the center is sqrt(4^2 + 3^2) = sqrt(25) = 5. Using the tangent length formula, the length of the tangent PA is sqrt(d^2 - r^2) = sqrt(5^2 - 3^2) = 4. In the right triangle OAP, the altitude from A to the line OP is given by (r * PA) / d = (3 * 4) / 5 = 12/5. The area of triangle PAB, where AB is the chord of contact, is (1/2) * AB * (altitude from P), but it is simpler to compute as the combined area of two identical right triangles formed by the line from P to the center, yielding Area = AB * (12/5) / 2 = (24/5) * (6/5) / 2 = 144/25. Wait, recalculating using the proper area formula for the triangle formed by the pair of tangents: Area = r * PA^2 / d = 3 * 16 / 5 = 48/5. Let's re-verify; Area = (r * PA / d^2) * PA = PA^2 * r / d^2 = 16 * 3 / 25 = 48/25? No, the correct formula for the area of triangle PAB is (1/2) * (length of chord of contact) * (distance from P to chord). Chord length = 2 * r * PA / d = 2 * 3 * 4 / 5 = 24/5. Distance from P to chord = PA^2 / d = 16 / 5. Area = (1/2) * (24/5) * (16/5) = 192/25 square units.