If the circle C1, x2 + y2 = 16 intersects another circle C2 of radius 5 in such a manner that the common chord is of maximum length and has a slope equal to (3/4), find the co-ordinates of centre C2.
Let the centre of the circle C2 is Q(h, k), Equation of the circle C2 is
or x2 + y2 – 2xh – 2yk + h2 + k2 – 25 = 0
And equation of circle
∴ Equation of common chord is
C1 – C2 = 0
or 2hx + 2ky – (h2 + k2 – 9) = 0 …(1)
Slope of this line = –h/k
But, it is given that its slope = 3/4
or 3k + 4h = 0 …(2)
Let P be the length perpendicular from P(0, 0) on chord (1), then
Length of this chord AB = 2AM
This chord has maximum length, then p = 0 then from (3),
h2 + k2 – 9 = 0 …(4)
Solving (2) and (4) we get
Find the equation of a circle which touches the x-axis and the line 4x – 3y+ 4 = 0. Its centre lies in the third quadrant and lies on the line x – y – 1 = 0.
Find the equations of the circle which passes through the origin and cut off chords of length a from each of the lines y = x and y = –x.
Determine the radius of the circle, two of whose tangents are the lines 2x+ 3y – 9 = 0 and 4x + 6y + 19 = 0.
Find the equation of the circle which touches the circle
x2 + y2 – 6x + 6y + 17 = 0 externally and to which the lines
x2 – 3xy – 3x + 9y = 0 are normals.
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Find the radius of smaller circle which touches the straight line 3x – y = 6 at (1, –3) and also touches the line y = x.
2x – y + 4 = 0 is a diameter of the circle which circumscribed a rectangleABCD. If the co-ordinates of A and B are A(4, 6) and B(1, 9), find the area of rectangle ABCD.
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