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
\(\frac{9}{2}\)
\(\frac{3\sqrt17}{2}\)
\(\frac{3\sqrt17}{4}\)
9
The Correct Option is B
Solution and Explanation
Let point P be (h, k)
\((ℎ–1)^2+(k–2)^2+(ℎ+2)^2+(k–1)^2=14\)
\(2ℎ^2+2k^2+2ℎ–6k–4=0\)
Locus of point P : x2 + y2 + x – 3y – 2 = 0
Intersection with x-axis,
x2 + x – 2 = 0
x = –2, 1
Intersection with y-axis,
y2 – 3y – 2 = 0
\(y=\frac{3±\sqrt17}{2}\)
Area of the quadrilateral ACBD is \(=\frac{1}{2}(|x1|+|x2|)(|y1|+|y2|)\)
\(=\frac{1}{2}×3×\sqrt17=\frac{3\sqrt17}{2}\)
So, the correct option is (B): \(\frac{3\sqrt17}{2}\)
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Concepts Used:
Coordinate Geometry
The central idea in coordinate geometry is to assign numerical coordinates to points in a plane or space, which allows us to describe their positions and relationships using algebraic equations. The most common coordinate system is the Cartesian coordinate system, named after the French mathematician and philosopher René Descartes.