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Find the equations of the circles whose centers lie on the line 4x + 3y = 2 and which touch the lines x + y + 4 = 0 and 7x - y + 4 = 0 There are 2 solutions to this question...

Find the equations of the circles whose centers lie on the line 4x + 3y = 2 and which touch the lines x + y + 4 = 0 and 7x - y + 4 = 0
 
There are 2 solutions to this question...

Grade:

3 Answers

Ravi
askIITians Faculty 69 Points
6 years ago
259-2055_Capture.PNG
This will give 2 values of theta. Hence, 2 values of coordinates of the centre. Use the length of prependicular to find the 2 corresponding radii. This should give the 2 eqns of circles satisfying the conditions mentioned.
Utkarsh
23 Points
6 years ago
Thank you so much Sir!!!
Shrutik
11 Points
3 years ago
Let the centre of the circle be C (h,k).
Since the Centre lies on the line 4x+3y=2,  we get 4h+3k=2
 
h = ( 2 – 3k ) / 4 …... Eqn (1) 
 
Since the lines 7x-y+4=0 and x+y+4 are touching the circles ( ie they are tangents ), let’s use the formula : Distance of point C(h,k) from these lines to get the radius as :
 
 \displaystyle \left | 7h-k+4 \right | / \sqrt{7*7+1*1}= \left | h+k+4 \right|/\sqrt{1*1+1*1} …...Eqn (2)
 
Substitute Eqn 1 in Eqn 2 to solve for k, giving k = -2 … thus h = 2 for +ve sign of Modulus and k = 6 and h = -4 for -ve sign of modulus.
 
Thus, the 2 centres are C1(2,-2)  and C2(-4,6). Radii of these 2 circles are 4/sqrt(2) and 6/sqrt(2) respectively.
Finally, using Centre -radius form, we get the 2  equations as 
x2 + y2 -4x+4y = 0 and x2+y2+8x-12y+34=0 
 
 
 

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