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.
Let the centre of circle is (–h, –k) where, h, k > 0. Since it touches x-axis
again (–h, –k) lies on x – y – 1 = 0 then
–h + k – 1 = 0 …(2)
Since the circle touches the line 4x – 3y + 4 = 0, so the length of perpendicular from (–h, –k) = radius of circle.
From (2), k = h + 1
∴ Required equation of circle is [from (1)]
If the origin be one limiting point of a system of co-axial circles of whichx2 + y2 + 3x + 4y + 25 = 0 is a member, find the other limiting point.
Find the radical axis of co-axial system of circles whose limiting points are (–1, 2) and (2, 3).
Find the equation of the circle which passes through the origin and belongs to the co-axial of circles whose limiting points are (1, 2) and (4, 3).
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 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.