If one of the diameters of the circle is a chord of another circle C, whose centre is the point of intersection of the lines 2x + 3y = 12 and 3x –2y = 5, then the radius of the circle C is : [2024]

Question

If one of the diameters of the circle $x^{2} + y^{2} - 10 x + 4 y + 13 = 0$ is a chord of another circle C, whose centre is the point of intersection of the lines 2x + 3y = 12 and 3x –2y = 5, then the radius of the circle C is : [2024]

Smart Achievers · Accepted Answer

(1)

Given, 2x + 3y = 12

3x – 2y = 5

Point of intersection = (3, 2)

$∴$ Centre is (3, 2)

$l = \sqrt{4 + 16} = 2 \sqrt{5}$

$l^{2} + R^{2} = r^{2}$

$\Rightarrow 20 + (25 + 4 - 13) = r^{2}$

$\Rightarrow 20 + 16 = r^{2} \Rightarrow r = 6$

Solution

Q.
If one of the diameters of the circle $x^{2} + y^{2} - 10 x + 4 y + 13 = 0$ is a chord of another circle C, whose centre is the point of intersection of the lines 2x + 3y = 12 and 3x –2y = 5, then the radius of the circle C is : [2024]

1 6

2 4

3 $\sqrt{20}$

4 $3 \sqrt{2}$

Ans.
(1)

Given, 2x + 3y = 12

3x – 2y = 5

Point of intersection = (3, 2)

$∴$ Centre is (3, 2)

$l = \sqrt{4 + 16} = 2 \sqrt{5}$

$l^{2} + R^{2} = r^{2}$

$\Rightarrow 20 + (25 + 4 - 13) = r^{2}$

$\Rightarrow 20 + 16 = r^{2} \Rightarrow r = 6$

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Solution

Q. If one of the diameters of the circle x2+y2–10x+4y+13=0 is a chord of another circle C, whose centre is the point of intersection of the lines 2x + 3y = 12 and 3x –2y = 5, then the radius of the circle C is : [2024]

1 6

2 4

3 20

4 32

Ans. (1) Given, 2x + 3y = 12 3x – 2y = 5 Point of intersection = (3, 2) ∴ Centre is (3, 2) l=4+16=25 l2+R2=r2 ⇒ 20+(25+4–13)=r2 ⇒ 20+16=r2 ⇒ r=6

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Q.
If one of the diameters of the circle $x^{2} + y^{2} - 10 x + 4 y + 13 = 0$ is a chord of another circle C, whose centre is the point of intersection of the lines 2x + 3y = 12 and 3x –2y = 5, then the radius of the circle C is : [2024]

3 $\sqrt{20}$

4 $3 \sqrt{2}$

Ans.
(1)

Given, 2x + 3y = 12

3x – 2y = 5

Point of intersection = (3, 2)

$∴$ Centre is (3, 2)

$l = \sqrt{4 + 16} = 2 \sqrt{5}$

$l^{2} + R^{2} = r^{2}$

$\Rightarrow 20 + (25 + 4 - 13) = r^{2}$

$\Rightarrow 20 + 16 = r^{2} \Rightarrow r = 6$