If the variable line 3x + 4y = a lies between the two circles (x – 1)2 + (y – 1)2 = 1 and (x – 9)2 + (y – 1)2 = 4, without intercepting a chord on either circle, then the sum of all the integral values of a is________.

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If the variable line 3x + 4y = a lies between the two circles (x – 1)² + (y – 1)² = 1 and (x – 9)² + (y – 1)² = 4, without intercepting a chord on either circle, then the sum of all the integral values of a is________.

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Raj Pandey | Contributor-Level 9

2 months ago

Since centres C₁ (1, 1) and C₂ (9, 1) lies opposite sides of the line3x + 4y = α

$((3 + 4 - α) ((27 + 4 - α) < 0 \Rightarrow α \in (7, 31) . . . . . . . . . . (i)$

Also length of perpendicular from centre of the circle is greater than radius of the circle.

$\frac{| 3 + 4 - α |}{5} \geq 1 \Rightarrow α \leq 2 o r α \geq 12 . . . . . . . . . . . . (i i)$

and $\frac{| 27 + 4 - α |}{5} \geq 2 \Rightarrow α \leq 21 o r α \geq 41 . . . . . . . . . . . (i i i)$

from (i), (ii) and (iii) we get $\in$ [12, 21]

$∴$ sum of all integers = 165

Since centres C1 (1, 1) and C2 (9, 1) lies opposite sides of the line3x + 4y = α <math> <mrow> <mo stretchy="false"> (</mo> <mo stretchy="false"> (</mo> <mn>3</mn> <mo>+</mo> <mn>4</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false"> (</mo> <mo stretchy="false"> (</mo> <mn>2</mn> <mn>7</mn> <mo>+</mo> <mn>4</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo><</mo> <mn>0</mn> <mo>⇒</mo> <mi>α</mi> <mo>∈</mo> <mrow> <mo> (</mo> <mrow> <mn>7</mn> <mo>, </mo> <mn>3</mn> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mrow> <mo> (</mo> <mrow> <mi>i</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> Also length of perpendicular from centre of the circle is greater than radius of the circle. <math> <mrow> <mfrac> <mrow> <mrow> <mo>|</mo> <mrow> <mn>3</mn> <mo>+</mo> <mn>4</mn> <mo>−</mo> <mi>α</mi> </mrow> <mo>|</mo> </mrow> </mrow> <mrow> <mn>5</mn> </mrow> </mfrac> <mo>≥</mo> <mn>1</mn> <mo>⇒</mo> <mi>α</mi> <mo>≤</mo> <mn>2</mn> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mi>o</mi> <mi>r</mi> <mtext> </mtext> <mtext> </mtext> <mi>α</mi> <mo>≥</mo> <mn>1</mn> <mn>2</mn> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mrow> <mo> (</mo> <mrow> <mi>i</mi> <mi>i</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> and <math> <mrow> <mfrac> <mrow> <mrow> <mo>|</mo> <mrow> <mn>2</mn> <mn>7</mn> <mo>+</mo> <mn>4</mn> <mo>−</mo> <mi>α</mi> </mrow> <mo>|</mo> </mrow> </mrow> <mrow> <mn>5</mn> </mrow> </mfrac> <mo>≥</mo> <mn>2</mn> <mo>⇒</mo> <mi>α</mi> <mo>≤</mo> <mn>2</mn> <mn>1</mn> <mtext> </mtext> <mtext> </mtext> <mi>o</mi> <mi>r</mi> <mtext> </mtext> <mtext> </mtext> <mi>α</mi> <mo>≥</mo> <mn>4</mn> <mn>1</mn> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mrow> <mo> (</mo> <mrow> <mi>i</mi> <mi>i</mi> <mi>i</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> from (i), (ii) and (iii) we get <math><mrow></mrow></math> <math> <mrow> <mo>∈</mo> </mrow> </math> [12, 21] <math> <mrow> <mo>∴</mo> </mrow> </math> sum of all integers = 165

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If the variable line 3x + 4y = a lies between the two circles (x – 1)² + (y – 1)² = 1 and (x – 9)² + (y – 1)² = 4, without intercepting a chord on either circle, then the sum of all the integral values of a is________.

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