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Let the circles S1 ≡ x2 + y2 – 4x – 8y + 4 = 0 and S2 be its image in the line y = x, the equation of the circle touching y = x at (1, 1) and orthogonal to S2 is  
  • a)
    x2 + y2  + x – 5y + 2 = 0  
  • b)
    x2 + y2 =  2  
  • c)
    x2 + y2  + x – 5y − 2 = 0  
  • d)
    (x – 3)2 + (y – 2)2 = 5  
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Let the circles S1 x2 + y2 4x 8y + 4 = 0 and S2 be its image in the...
Centre of circle S1 = (2, 4)  
Centre of circle S2 = (4, 2)  
Radius of circle S1 = radius of circle S2 = 4
∴ equation of circle S2  
(x – 4)2 + (y – 2)2 = 16  
⇒ x2 + y2 – 8x – 4y + 4 = 0  . .  . (i)
Equation of circle touching y = x at (1, 1) can be taken as  
(x – 1)2 + (y – 1)2 + λ(x – y) = 0
or, x2 + y2  + x (λ – 2) + y(– λ – 2) + 2 = 0  . . . (ii)  
As this is orthogonal to S2
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Most Upvoted Answer
Let the circles S1 x2 + y2 4x 8y + 4 = 0 and S2 be its image in the...
Given:
Equation of circle S1: x^2 + y^2 + 4x + 8y + 4 = 0
Equation of line y = x
Point of tangency on y = x: (1, 1)

We need to find the equation of the circle that touches the line y = x at (1, 1) and is orthogonal to the image of circle S1 in the line y = x.

Approach:
To find the equation of the circle orthogonal to S2, we need to find the center and radius of S2. Then we can use the distance formula to find the center of the required circle. The radius of the required circle will be the same as that of S2.

Solution:

Finding the Image of S1 (S2) in the line y = x:
1. Substitute y = x in the equation of S1:
x^2 + x^2 + 4x + 8x + 4 = 0
2x^2 + 12x + 4 = 0
Simplifying, we get:
x^2 + 6x + 2 = 0

2. The image of S1 in the line y = x is given by the equation x^2 + 6x + 2 = 0, which we will call S2.

Finding the Center and Radius of S2:
1. To find the center of S2, we need to find the x-coordinate of the vertex of the quadratic equation x^2 + 6x + 2 = 0.
The x-coordinate of the vertex can be found using the formula: x = -b/2a.
For the given equation, a = 1 and b = 6.
x = -6/2 = -3.

2. Substitute x = -3 in the equation x^2 + 6x + 2 = 0 to find the y-coordinate of the center of S2.
(-3)^2 + 6(-3) + 2 = 9 - 18 + 2 = -7.

3. The center of S2 is (-3, -7).

4. To find the radius of S2, substitute the coordinates of the center into the equation of S2 and solve for the radius.
(-3)^2 + 6(-3) + 2 = 9 - 18 + 2 = -7.
The radius of S2 is the square root of 7.

Finding the Center and Radius of the Required Circle:
1. The required circle touches the line y = x at (1, 1), which means the center lies on the line y = x.

2. Since the center of the required circle lies on the line y = x, the x-coordinate and y-coordinate of the center will be the same.

3. Let the x-coordinate and y-coordinate of the center be c.

4. The distance between the center of the required circle and the point of tangency (1, 1) is equal to the radius of S2.

5. Using the distance formula, we have:
sqrt((c - 1)^2 + (c - 1)^2) = sqrt(
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Let the circles S1 x2 + y2 4x 8y + 4 = 0 and S2 be its image in the line y = x, the equation of the circle touching y = x at (1, 1) and orthogonal to S2 is a)x2 + y2 + x 5y + 2 = 0 b)x2 + y2 = 2 c)x2 + y2 + x 5y 2 = 0 d)(x 3)2 + (y 2)2 = 5 Correct answer is option 'A'. Can you explain this answer?
Question Description
Let the circles S1 x2 + y2 4x 8y + 4 = 0 and S2 be its image in the line y = x, the equation of the circle touching y = x at (1, 1) and orthogonal to S2 is a)x2 + y2 + x 5y + 2 = 0 b)x2 + y2 = 2 c)x2 + y2 + x 5y 2 = 0 d)(x 3)2 + (y 2)2 = 5 Correct answer is option 'A'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let the circles S1 x2 + y2 4x 8y + 4 = 0 and S2 be its image in the line y = x, the equation of the circle touching y = x at (1, 1) and orthogonal to S2 is a)x2 + y2 + x 5y + 2 = 0 b)x2 + y2 = 2 c)x2 + y2 + x 5y 2 = 0 d)(x 3)2 + (y 2)2 = 5 Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let the circles S1 x2 + y2 4x 8y + 4 = 0 and S2 be its image in the line y = x, the equation of the circle touching y = x at (1, 1) and orthogonal to S2 is a)x2 + y2 + x 5y + 2 = 0 b)x2 + y2 = 2 c)x2 + y2 + x 5y 2 = 0 d)(x 3)2 + (y 2)2 = 5 Correct answer is option 'A'. Can you explain this answer?.
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