Let A(a, 0), B(b, 2b + 1) and C(0, b), b $\neq$ 0, $| b | \neq 1,$ be points such that the area of triangle ABC is 1 sq. unit, then the sum of all possible values of a is

Option 1 -

$\frac{- 2 b^{2}}{b + 1}$

Option 2 -

$\frac{2 b}{b + 1}$

Option 3 -

$\frac{2 b^{2}}{b + 1}$

Option 4 -

$\frac{- 2 b}{b + 1}$

3 Views | Posted 2 months ago

Asked by Shiksha User

1 Answer

Answered by

alok kumar singh | Contributor-Level 10

2 months ago

Correct Option - 1

Detailed Solution:

$a r (Δ A B C) = \frac{1}{2} | \begin{matrix} a & 0 & 1 \\ b & 2 b + 1 & 1 \\ 0 & b & 1 \end{matrix} | = \frac{1}{2} [a (2 b + 1 - b) + (b^{2})] = \frac{1}{2} [a b + a + b^{2}]$

Given ar ( $Δ A B C)$ = 1 so |ab + a + b²| = 2

$\Rightarrow a (b + 1) + b^{2} = \pm 2$

a = $\frac{- b^{2} + 2}{b + 1}, \frac{- b^{2} - 2}{b + 1}$

So sum = $\frac{- 2 b^{2}}{b + 1}$

<math> <mrow> <mi>a</mi> <mi>r</mi> <mrow> <mo>(</mo> <mrow> <mi>Δ</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mrow> <mo>|</mo> <mrow> <mtable> <mtr columnalign="center"> <mtd columnalign="center"> <mi>a</mi> </mtd> <mtd columnalign="center"> <mn>0</mn> </mtd> <mtd columnalign="center"> <mn>1</mn> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mi>b</mi> </mtd> <mtd columnalign="center"> <mn>2</mn> <mi>b</mi> <mo>+</mo> <mn>1</mn> </mtd> <mtd columnalign="center"> <mn>1</mn> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mn>0</mn> </mtd> <mtd columnalign="center"> <mi>b</mi> </mtd> <mtd columnalign="center"> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mrow> <mo>[</mo> <mrow> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>b</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mi>b</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mrow> <mo>[</mo> <mrow> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mo>+</mo> <msup> <mrow> <mi>b</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mrow> </math>  Given ar (  <math> <mrow> <mi>Δ</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> </math> = 1 so |ab + a + b2| = 2  <math> <mrow> <mo>⇒</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mrow> <mi>b</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>±</mo> <mn>2</mn> </mrow> </math>             a = <math> <mrow> <mfrac> <mrow> <mo>−</mo> <msup> <mrow> <mi>b</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> </mrow> <mrow> <mi>b</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mo>,</mo> <mfrac> <mrow> <mo>−</mo> <msup> <mrow> <mi>b</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <mi>b</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </math>  So sum = <math> <mrow> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <msup> <mrow> <mi>b</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>b</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </math>

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Let A(a, 0), B(b, 2b + 1) and C(0, b), b $\neq$ 0, $| b | \neq 1,$ be points such that the area of triangle ABC is 1 sq. unit, then the sum of all possible values of a is

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Let A(a, 0), B(b, 2b + 1) and C(0, b), b ≠ 0, | b | ≠ 1 , be points such that the area of triangle ABC is 1 sq. unit, then the sum of all possible values of a is

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Let A(a, 0), B(b, 2b + 1) and C(0, b), b $\neq$ 0, $| b | \neq 1,$ be points such that the area of triangle ABC is 1 sq. unit, then the sum of all possible values of a is