SATHEE: Circle 1 Question 4

Circle 1 Question 4

4. A square is inscribed in the circle $x^{2} + y^{2} - 6 x + 8 y - 103 = 0$ with its sides parallel to the coordinate axes. Then, the distance of the vertex of this square which is nearest to the origin is

(2019 Main, 11 Jan II)

(a) 6

(b) 13

(d) $\sqrt{137}$

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Answer:

Correct Answer: 4. (c)

Solution:

Given equation of circle is $x^{2} + y^{2} - 6 x + 8 y - 103 = 0$ , which can be written as $(x - 3)^{2} + (y + 4)^{2} = 128 = (8 \sqrt{2})^{2}$

$∴$ Centre $= (3, - 4)$ and radius $= 8 \sqrt{2}$

Now, according to given information, we have the following figure.

For the coordinates of $A$ and $C$ .

Consider, $\frac{x - 3}{\frac{1}{\sqrt{2}}} = \frac{y + 4}{\frac{1}{\sqrt{2}}} = \pm 8 \sqrt{2}$

[using distance (parametric) form of line,

$\frac{x - x_{1}}{\cos θ} = \frac{y - y_{1}}{\sin θ} = r]$

$\Rightarrow x = 3 \pm 8, y = - 4 \pm 8$

$∴ A (- 5, - 12)$ and $C (11, 4)$

Similarly, for the coordinates of $B$ and $D$ , consider

$\begin{aligned} \frac{x - 3}{- \frac{1}{\sqrt{2}}} = \frac{y + 4}{\frac{1}{\sqrt{2}}} = \pm 8 \sqrt{2} [in this case, θ = 135^{\circ} \\ \Rightarrow x = 3 \mp 8, y = - 4 \pm 8 \\ ∴ B (11, - 12) and D (- 5, 4) \\ O B = \sqrt{121 + 144} = \sqrt{265} \\ O C = \sqrt{121 + 16} = \sqrt{137} \\ O D = \sqrt{25 + 16} = \sqrt{41} \end{aligned}$

Circle 1 Question 4

4. A square is inscribed in the circle $x^{2} + y^{2} - 6 x + 8 y - 103 = 0$ with its sides parallel to the coordinate axes. Then, the distance of the vertex of this square which is nearest to the origin is

Answer:

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Circle 1 Question 4

4. A square is inscribed in the circle x2+y2−6x+8y−103=0 with its sides parallel to the coordinate axes. Then, the distance of the vertex of this square which is nearest to the origin is

Answer:

Engineering Preparation

Medical Preparation

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Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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4. A square is inscribed in the circle $x^{2} + y^{2} - 6 x + 8 y - 103 = 0$ with its sides parallel to the coordinate axes. Then, the distance of the vertex of this square which is nearest to the origin is