SATHEE: Circle 5 Question 23

Circle 5 Question 23

23. Let $A$ be the centre of the circle $x^{2} + y^{2} - 2 x - 4 y - 20 = 0$ . Suppose that, the tangents at the points $B (1, 7)$ and $D$ $(4, - 2)$ on the circle meet at the point $C$ . Find the area of the quadrilateral $A B C D$ .

$(1981, 4 M)$

Integer Answer Type Question

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Answer:

Correct Answer: 23. 75 sq units

Solution:

Equation of tangents at $(1, 7)$ and $(4, - 2)$ are

$\begin{array}{lcrl} x + 7 y - (x + 1) - 2 (y + 7) - 20 & = 0 \\ \Rightarrow & 5 y = 35 \Rightarrow & y = 7 \\ and & 4 x - 2 y - (x + 4) - 2 (y - 2) - 20 & = 0 \\ \Rightarrow & 3 x - 4 y & = 20 \end{array}$

$∴$ Point $C$ is $(16, 7)$ .

$∴$ Vertices of a quadrilateral are

$A (1, 2), B (1, 7), C (16, 7), D (4, - 2)$

$∴$ Area of quadrilateral $A B C D$

$\begin{aligned} = Area of △ A B C + Area of △ A C D \\ = \frac{1}{2} \times 15 \times 5 + \frac{1}{2} \times 15 \times 5 = 75 sq units \end{aligned}$

Circle 5 Question 23

23. Let $A$ be the centre of the circle $x^{2} + y^{2} - 2 x - 4 y - 20 = 0$ . Suppose that, the tangents at the points $B (1, 7)$ and $D$ $(4, - 2)$ on the circle meet at the point $C$ . Find the area of the quadrilateral $A B C D$ .

Integer Answer Type Question

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Circle 5 Question 23

23. Let A be the centre of the circle x2+y2−2x−4y−20=0. Suppose that, the tangents at the points B(1,7) and D (4,−2) on the circle meet at the point C. Find the area of the quadrilateral ABCD.

Integer Answer Type Question

Answer:

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23. Let $A$ be the centre of the circle $x^{2} + y^{2} - 2 x - 4 y - 20 = 0$ . Suppose that, the tangents at the points $B (1, 7)$ and $D$ $(4, - 2)$ on the circle meet at the point $C$ . Find the area of the quadrilateral $A B C D$ .