Advertisements
Advertisements
Question
Two circles of radii 5cm and 3cm with centres O and P touch each other internally. If the perpendicular bisector of the line segment OP meets the circumference of the larger circle at A and B, find the length of AB.
Solution
OA = OQ = 5 cm (Radius of bigger circle)
PQ = 3cm (Radius of smaller circle)
OP = 2cm
Perpendicular bisector of OP, i.e. AB meets OP at M.
OM = MP = `1/2` OP = 1 cm
In right Δ OMA,
By Pythagoras theorem,
OA2 = OM2 + MA2
MA2 = 52 -12
= 25 - 1
= 24
AM = `2 sqrt 6` cm
AM = MB = `2 sqrt 6` cm
AB = AM + MB = `2 sqrt 6 + 2 sqrt 6 = 4 sqrt 6`
APPEARS IN
RELATED QUESTIONS
Two circle touch each other internally. Show that the tangents drawn to the two circles from any point on the common tangent are equal in length.
In the given figure, two circles touch each other externally at point P. AB is the direct common tangent of these circles. Prove that :
- tangent at point P bisects AB,
- angles APB = 90°.
Two circle with centres O and O' are drawn to intersect each other at points A and B. Centre O of one circle lies on the circumference of the other circle and CD is drawn tangent to the circle with centre O' at A. Prove that OA bisects angle BAC.
Two circles intersect each other at points A and B. A straight line PAQ cuts the circles at P and Q. If the tangents at P and Q intersect at point T; show that the points P, B, Q and T are concyclic.
Two circles intersect each other at points A and B. Their common tangent touches the circles at points P and Q as shown in the figure. Show that the angles PAQ and PBQ are supplementary.
Two circles intersect each other at points C and D. Their common tangent AB touches the circles at point A and B. Prove that :
∠ ADB + ∠ ACB = 180°
Two circles with centres O and P intersect each other at A and B as shown in following fig. Two straight lines MAN and RBQ are drawn parallel to OP.
Prove that (i) MN = 20 P (ii) MN= RQ.
Radii of two circles are 6.3 cm and 3.6 cm. State the distance between their centers if -
they touch each other internally.
P and Q are the centre of circles of radius 9 cm and 2 cm respectively; PQ = 17 cm. R is the centre of circle of radius x cm, which touches the above circles externally, given that ∠ PRQ = 90°. Write an equation in x and solve it.
Two circles with centres O and O' touch each other at point L. Prove that, a tangent through L bisects the common tangent AB of the two circles at point M.
Given: AB is a common tangent of the two circles that touch each other at point L. ML is a tangent through point L.
To prove: M is a mid-point of the tangent AB or MA = MB.
Proof: From the figure,
M is an external point that draws two tangents, MA and ML to the circle with the centre O.
So, `square` = `square` ......(i)
Similarly, M draws two tangents ML and MB to the circle with the centre O'.
So, `square` = `square` ......(ii)
From the equations (i) and (ii),
`square` = `square`
Hence, the tangent at the point L, bisects the common tangent, AB of the two circles at point M.
