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.
x2 + y2 – 6x – 2y + 1 = 0
The given circle is
x2 + y2 – 6x + 6y + 17 = 0
centre and radius of this circle are (3, –3) and
Respectively. But given the required circle has normals
x2 – 3xy – 3x + 9y = 0
or (x – 3) (x – 3y) = 0
or x = 3 and x – 3y = 0 …(1)
But point of intersection of normals is the centre of the circle.
Point of intersection of normals represented by (1) is (3, 1) which is centre of the required circle. Since given circle and required circle touch each other externally, then (if radius of required circles is r)
∴ Equation of required circle is
(x – 3)2 + (y – 1)2 = 32
or x2 + y2 – 6x – 2y + 1 = 0
Find the equation of the image of the circle x2 + y2 + 16x – 24y + 183 = 0 by the line mirror 4x + 7y + 13 = 0.
Find the area of the triangle formed by the tangents drawn from the point (4, 6) to the circle x2 + y2 = 25 and their chord of contact. Also find the length of chord of contact.
Find the lengths of external and internal common tangents to two circlesx2 + y2 + 14x – 4y + 28 = 0 and x2 + y2 – 14x + 4y – 28 = 0.
Find the lengths of common tangents of the circles x2 + y2 = 6x and x2 +y2 + 2x = 0.
Find the equation of the circle circumscribing the triangle formed by the lines:
x + y = 6, 2x + y = 4 and x + 2y = 5,
Without finding the vertices of the triangle.
Find the equation of the circle circumscribing the quadrilateral formed by the lines in order are 5x + 3y – 9 = 0, x – 3y = 0, 2x – y = 0, x + 4y – 2 = 0 without finding the vertices of quadrilateral.
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 a circle which passes through the point
(2, 0) and whose centre is the limit of the point of intersection of the lines 3x + 5y = 1and (2 + c)x + 5c2y = 1as c → 1.