Question

Let $A$ be the set of points in the $XY$-plane which are equidistant from $P(-1, 0)$ and $Q(1, 0)$. Let $B$ be the set of points in the $XY$-plane which are equidistant from $A$ and $Q$. If $(5, y)$ is a point in $B$, then what is the value of $y^2$?

• A. 9
• B. 1
• C. 4
• D. 16

Hint:

The definition of set \(B\) is equivalent to the definition of a parabola.
The line \(x = 0\) acts as the directrix, and the point \(Q(1, 0)\) acts as the focus.
Remember that the standard equation of a parabola with focus \((a, 0)\) and directrix \(x = -a\) is \(y^2 = 4ax\).
Shifting the coordinates appropriately gives \(y^2 = 2x - 1\) directly.

Answer 1

The Correct Option is A

•

Step 1 : Understanding the Question:
In this question, we need to find the locus of two sets of points.
First, the set \(A\) is defined as the set of points equidistant from two fixed points \(P(-1, 0)\) and \(Q(1, 0)\).
Second, the set \(B\) is the set of points equidistant from the set \(A\) (which will be a line) and the point \(Q(1, 0)\).
Once the equation representing set \(B\) is obtained, we substitute \(x = 5\) to find the value of \(y^2\).

•

Step 2 : Key Formula or Approach:
The locus of points equidistant from two fixed points is the perpendicular bisector of the line segment joining them.
The distance from a point \((x_0, y_0)\) to a line \(ax + by + c = 0\) is given by:
\[ d = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}} \] The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Equating the two distances gives the locus of set \(B\).

•

Step 3 : Detailed Explanation:
Let us find the set \(A\) first.
The points \(P(-1, 0)\) and \(Q(1, 0)\) lie on the \(X\)-axis.
The midpoint of the segment \(PQ\) is:
\[ M = \left( \frac{-1 + 1}{2}, \frac{0 + 0}{2} \right) = (0, 0) \] Since the segment \(PQ\) lies on the \(X\)-axis, its perpendicular bisector is the \(Y\)-axis.
Thus, set \(A\) is the line with the equation:
\[ x = 0 \] Now, let us find the set \(B\).
Let a point in the set \(B\) be denoted by \((x, y)\).
The distance from \((x, y)\) to the line \(A\) (the \(Y\)-axis, \(x = 0\)) is:
\[ d_1 = |x| \] The distance from \((x, y)\) to the point \(Q(1, 0)\) is:
\[ d_2 = \sqrt{(x - 1)^2 + (y - 0)^2} = \sqrt{(x - 1)^2 + y^2} \] Since the points in \(B\) are equidistant from \(A\) and \(Q\)., we have:
\[ d_1 = d_2 \implies |x| = \sqrt{(x - 1)^2 + y^2} \] Squaring both sides of the equation:
\[ x^2 = (x - 1)^2 + y^2 \] \[ x^2 = x^2 - 2x + 1 + y^2 \] \[ 0 = -2x + 1 + y^2 \implies y^2 = 2x - 1 \] This equation represents a parabola.
We are given that the point \((5, y)\) lies in the set \(B\).
Substituting \(x = 5\) into the equation of the locus \(B\):
\[ y^2 = 2(5) - 1 \] \[ y^2 = 10 - 1 = 9 \] Therefore, the value of \(y^2\) is 9.

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Step 4 : Final Answer:
The value of \(y^2\) is 9.
This corresponds to option (A).