SHEET-4 INTERSECTION OF TWO CIRCLES,
CHORD OF CONTACT, COMMON TANGENTS
1. Examine whether the two circles x2 + y2 – 2x – 4y = 0 and x2 + y2 – 8y – 4 = 0 touch each other
internally or externally.
2. Prove that the circle x2 + y2 + 2ax + c2 = 0 and x2 + y2 + 2by + c2 = 0 touch each other if
1 1 1
2 + 2 = 2 .
a b c
3. Prove that the circle x2 + y2 + 2x + 2y + 1 = 0 and x2 + y2 – 4x – 6y – 3 = 0 touch each other and find
the coordinates of the point of contact.
4. If the two circles x2 + y2 – 8x + 2y + 8 = 0 and (x – 1)2 + (y – 3)2 = r2 intersect in two distinct points,
then prove that 2 < r < 8.
5. Find the locus of the centres of the circles, which touch the two circles x2 + y2 = a2 and x2 + y2 – 4ax = 0 and
are external to both.
6. Determine the position of the circles
x2 + y2 + 2x – 8y + 13 = 0
x2 + y2 – 12x – 14y + 76 = 0
and find the equation of the common tangents.
7. Find the coordinates of the point at which the circles x 2 + y2 – 4x – 2y – 4 = 0 and
x2 + y2 – 12x – 8y – 12 = 0 touch each other. Also, find the equations of common tangents touching the
circle in distinct points.
8. Two circles, each of radius 5 units, touch each other at (1, 2). If the equation of their common tangent
is 4x + 3y = 10, find the equation of the circle.
9. Show that the common tangents to the circle x2 + y2 – 6x = 0 and x2 + y2 + 2x = 0 form an equilateral
triangle.
10. Find all the common tangents to the circles x2 + y2 – 2x – 6y + 9 = 0 and x2 + y2 + 6x – 2y + 1 = 0.
11. Find the middle point of the chord intercepted on line lx + my + n = 0 by the circle x2 + y2 = a2.
12. Through a fixed point (h, k), secants are drawn to the circle x2 + y2 = r2. Show that the locus of mid-point
of the portions of secants intercepted by the circle is x2 + y2 = hx + ky.
13. The chord of contact of the circle x2 + y2 = b2 is generated by a point on the circle x2 + y2 = a2 and the chord
of contact touches the circle x2 + y2 = c2. Prove that a, b, c are in G.P.
14. Find the equation of the chord of x2 + y2 – 6x + 10y – 9 = 0 which is bisected at (–2, 4).
15. Find the condition that chord of contact of any external point (h, k) to the circle x2 + y2 = a2 should subtend
right angle at the centre of the circle.
16. If the two circles, x2 + y2 + 2g1x + 2f1y = 0 and x2 + y2 + 2g2x + 2f2y = 0 touches each other, then -Prove that
f1g2 = f2g
1
FIITJEE Ltd., ICES House, 29 – A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 26515949, 26569493, Fax:011-
26513 942.
SHEET-4 INTERSECTION OF TWO CIRCLES,
CHORD OF CONTACT, COMMON TANGENTS
2 1
1. internally 2. , 5. 12x2 – 4y2 – 24ax + 9a2 = 0
5 5
26 21 5 33 x 9 21 57
6. y– = y+2= (x + 15)
5 24 5 48
2 4
7. , No direct common tangent
5 5
8. (x – 5)2 + (y – 5)2 = 25 and (x + 3)2 + (y + 1)2 = 25
10. x = 0, 3x + 4y – 10 0, y = 4, 4x – 3y = 0
n nm
11. 2 m2 , 2 m2
14. 5x – 9y + 46 = 0
15. h2 + k2 = 2a2
FIITJEE Ltd., ICES House, 29 – A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 26515949, 26569493, Fax:011-
26513 942.
FIITJEE Ltd., ICES House, 29 – A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 26515949, 26569493, Fax:011-
26513 942.
FIITJEE Ltd., ICES House, 29 – A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 26515949, 26569493, Fax:011-
26513 942.
FIITJEE Ltd., ICES House, 29 – A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 26515949, 26569493, Fax:011-
26513 942.
FIITJEE Ltd., ICES House, 29 – A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 26515949, 26569493, Fax:011-
26513 942.
