Question

Two circles touch internally. Sum of their areas is 116π cm sq and the distance between their centres is 6cm. Find the radio of circles

Answer 1

Let a circle with center O And radius R.letanother circle inside the first circle  with center o' and radius r .A/Q,Area of 1st circle + area of 2nd circle = 116π cm²⇒ πR² + πr² = 116π ⇒ π(R² + r²)  = 116π⇒ R² + r² =116 --------------------(i)Now,Distance between the centers of circles = 6 cmi.e, R - r = 6⇒  R = r + 6  -------------------(ii)From Eqn (i) & (ii),(r + 6)² + r² = 116⇒ r² + 12r +36 + r² =116⇒ 2r² +12r +36 -116 = 0⇒ 2r² +12r - 80 = 0⇒ r² +6r - 40 = 0⇒ r² +10r - 4r - 40 = 0⇒ r(r + 10) - 4(r + 10) = 0⇒ (r + 10)(r - 4) = 0 hence  r = 4 cmr ≠ -10 cm        {∵ length can't be -ve}Therefore radii of the circles are r = 4 cm ,R = 4 + 6 = 10 cm.

Answer 2

The circles touch internally.

$\sf \:\:\:\:\therefore\:\: difference\: of \:their\: radii\:$$\sf \:\:\:\:=\: distance\: between\: their \:centres\:=\:6\:cm$

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$\orange\longrightarrow$Let the radii of given circles be r cm and (r + 6) cm

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$\bf\sf \:\:Sum\: of\: their \:areas= [\pi r²+\pi(r+6)²]cm²$

$\bf\sf \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\pi[r²+(r+6)²]cm²$

$\:\:\:\:\:\:\:\:\:\:\:──────────────────$

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$\sf \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\therefore\:\:\pi[r²+(r+6)²]=116\pi$

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$\sf \:\:\:\:\:\:\:\:\:\:\:\: \longmapsto\:r²+(r+6)²=116$

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$\sf \:\:\:\: \longmapsto\:2r²+12r-80=0$

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$\sf \:\:\:\:\:\:\:\:\:\longmapsto\:r²+6r-40=0$

$\:\:$

$\sf \:\:\:\:\:\:\:\:\:\:\:\:\:\longmapsto\:(r+10)(r-4)=0$

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$\sf \:\:\:\:\:\:\:\:\:\:\:\:\:\longmapsto\:r+10=0\:\:or\:\:r-4=0$

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$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$\underline{\boxed{\sf{r=\frak{\red{4}}}}}$

$\:\:\:\underline{\boxed{\sf{r=\frak{\pink{-10\:[neglect,\:as\:r\:can't\:be\: negative]}}}}}$

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${\bold { \underline{\small{ \therefore\:the \:radii\: of \: given \:circles\: are\:-}}}}$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\sf \bullet\: 4\:cm$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\sf \bullet\: 10\:cm$

$\:\:\:\:━━━━━━━━━━━━━━━━━━━━━━━━$

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More to know -

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${\normalsize{\frak{\pmb{\underline{ When\: two \:circles \:touch\: internally,}}}}}$$\sf \:\:\:|fig.(i)|$

• $\sf Distance\: between\: their\: centres= difference \:of \:their\: radii$

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${\normalsize{\frak{\pmb{\underline{ When\:two \:circles \:touch\: externally,}}}}}$$\sf \:|fig.(ii)|$

• $\sf Distance\: between\: their\: centres= sum \:of \:their\: radii$