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प्रश्न
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.
उत्तर
From Q, QA and QP are two tangents to the circle with centre O
Therefore, QA = QP ...(i)
Similarly, from Q, QB and QP are two tangents to the circle with centre O'
Therefore, QB = QP ...(ii)
From (i) and (ii)
QA = QB
Therefore, tangents QA and QB are equal.
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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.
