A point P moves so that the sum of squares of its distances from the points (1, 2) and (-2, 1) is 14. Let f(x, y) = 0 be the locus of P, which intersects the x-axis at the points A , B and the y-axis at the points C, D. Then the area of the quadrilateral ACBD is equal to:

Option 1 -

$\frac{9}{2}$

Option 2 -

$\frac{3 \sqrt[]{17}}{2}$

Option 3 -

$\frac{3 \sqrt[]{17}}{4}$

Option 4 -

2 Views | Posted 2 weeks ago

Asked by Shiksha User

1 Answer

Answered by

Raj Pandey | Contributor-Level 9

2 weeks ago

Correct Option - 2

Detailed Solution:

Let point P : (h, k)

Therefore according to question, ${(h - 1)}^{2} + {(k - 2)}^{2} + {(h + 2)}^{2} + {(k - 1)}^{2} = 14$

$∴$ locus of P(h, k) is $x^{2} + y^{2} + x - 3 y - 2 = 0$

Now intersection with x – axis are $x^{2} + x - 2 = 0 \Rightarrow x = - 2, 1$

Now intersection with y – axis are $y^{2} - 3 y - 2 = 0 \Rightarrow y = \frac{3 \pm \sqrt[]{17}}{2}$

Therefore are of the quadrilateral ABCD is = $\frac{1}{2} (| x_{1} | + | x_{2} |) (| y_{1} | + | y_{2} |) = \frac{1}{2} \times 3 \times \sqrt[]{17} = \frac{3 \sqrt[]{17}}{2}$

Let point P : (h, k)Therefore according to question,  <math> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mi>h</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mi>h</mi> <mo>+</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mn>4</mn> </mrow> </math> <math> <mrow> <mo>∴</mo> </mrow> </math> locus of P(h, k) is <math> <mrow> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mi>y</mi> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>0</mn> </mrow> </math>  Now intersection with x – axis are  <math> <mrow> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>0</mn> <mo>⇒</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </math>  Now intersection with y – axis are  <math> <mrow> <msup> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>y</mi> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>0</mn> <mo>⇒</mo> <mi>y</mi> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mo>±</mo> <mroot> <mrow> <mn>1</mn> <mn>7</mn> </mrow> <mrow></mrow> </mroot> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math>  Therefore are of the quadrilateral ABCD is =  <math> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mrow> <mo>|</mo> <mrow> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mo>|</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mo>|</mo> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mo>|</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>×</mo> <mn>3</mn> <mo>×</mo> <mroot> <mrow> <mn>1</mn> <mn>7</mn> </mrow> <mrow></mrow> </mroot> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mroot> <mrow> <mn>1</mn> <mn>7</mn> </mrow> <mrow></mrow> </mroot> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math>

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A point P moves so that the sum of squares of its distances from the points (1, 2) and (-2, 1) is 14. Let f(x, y) = 0 be the locus of P, which intersects the x-axis at the points A , B and the y-axis at the points C, D. Then the area of the quadrilateral ACBD is equal to:

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