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Q) The coordinates of the centre of a circle are (x − 7, 2 x). Find the value(s) of ’x’, if the circle passes through the point (− 9, 11) and has radius 5 √2 units.
(Q23 – 30/1/1 – CBSE 2026 Question Paper)
Ans:
Step 1: We are given, Coordinates of the centre = (x – 7, 2x)
Coordinates of the point on circle = (- 9, 11)
and Radius of the circle = 5√2 units
Step 2: Distance between 2 points by distance formula:
D = 
∴ Distance between centre (x – 7, 2x) and point (- 9, 11)
= 
= 
= 
= 
= 
= 
………. (i)
Step 3: ∵ This distance between center and the point on the circle is radius and it is given that radius length = 5√2
∴ = 5√2
By squaring on both sides:
∴ 
∴ (5 x 2 + 125 – 40 x) = 50
∴ 5 x 2 – 40 x + 75 = 0
∴ x 2 – 8 x + 15 = 0
By Mid-term splitting, we get:
∴ x 2 – 5 x – 3 x + 15 = 0
∴ x (x – 5) – 3 (x – 5) = 0
∴ (x – 3) (x – 5) = 0
∴ x = 3 and x = 5
Therefore, the values of x are 3 and 5.
Check: Let’s check if the values of x are correct or not:
For x = 3, Length of radius from equation (i) = √(5 (3)2 + 125 – 40 (3)) = √(45 + 125 – 120) = √50 = 5√2 units
It matches the given condition, hence our answer x = 3 is correct.
For x = 5, Length of radius from equation (i) = √(5 (5)2 + 125 – 40 (5)) = √(125 + 125 – 200) = √50 = 5√2 units
It also matches the given condition, hence our answer x = 5 is correct.
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