Two circles each of the radius 5 units touch each other at the point (1, 2). If the equation of their common tangent is 4x + 3y = 10, and C₁ (a, b) and C_{2 $\left(\gamma ,\delta \right),C_1\ne C_2$}  are their centres, then $\left|\left(\alpha +\beta \right)\left(\gamma +\delta \right)\right|$ is equal to______

 Slope of C₁C₂ = $\frac{3}{4}=tan\theta$  

By parametric form C₁ (1 + 5 cosθ, 2 + 5sinθ)

& C₂ (1 – 5 cos θ, 2 – 5 sinθ)

C_{1 $\left(1+5\times \frac{4}{5}.2+5\times \frac{3}{5}\right)\&C_2\left(1-5\times \frac{4}{5}.2-5\times \frac{3}{5}\right)$}

So, $\left|\left(\alpha +\beta \right)\left(r+\delta \right)\right|=40$  

 

<p>&nbsp;Slope of C₁C₂ =<span contenteditable="false"> <math> <mrow> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> <mo>=</mo> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mi>θ</mi> </mrow> </math> </span>&nbsp;&lt;!- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1815988090"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;</p><p>By parametric form C₁ (1 + 5 cosθ, 2 + 5sinθ)</p><p>&amp; C₂ (1 – 5 cos θ, 2 – 5 sinθ)</p><p>C_{1<span contenteditable="false"> <math> <mrow> <mrow> <mo> (</mo> <mrow> <mn>1</mn> <mo>+</mo> <mn>5</mn> <mo>×</mo> <mfrac> <mrow> <mn>4</mn> </mrow> <mrow> <mn>5</mn> </mrow> </mfrac> <mo>.</mo> <mn>2</mn> <mo>+</mo> <mn>5</mn> <mo>×</mo> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>5</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>&amp;</mo> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mrow> <mo> (</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>5</mn> <mo>×</mo> <mfrac> <mrow> <mn>4</mn> </mrow> <mrow> <mn>5</mn> </mrow> </mfrac> <mo>.</mo> <mn>2</mn> <mo>−</mo> <mn>5</mn> <mo>×</mo> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>5</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </math> </span>} &lt;!- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1815988091"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;</p><p>So, <span contenteditable="false"> <math> <mrow> <mrow> <mo>|</mo> <mrow> <mrow> <mo> (</mo> <mrow> <mi>α</mi> <mo>+</mo> <mi>β</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo> (</mo> <mrow> <mi>r</mi> <mo>+</mo> <mi>δ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>4</mn> <mn>0</mn> </mrow> </math> </span>&nbsp;&lt;!- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1815988092"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;</p><div class="photo-widget-full"><div class="figure"><picture><source srcset="" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1754462319phpZdEjy9.jpeg" alt="" width="252" height="159" loading="lazy"></picture></div></div><p>&nbsp;</p>

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