A line through is drawn parallel to . Point moves such that its distances from the line and the vertex are equal. If locus of cuts at and and at , then area of is [2006]

Question

$A B C D$ is a square of side length 2 units. $C_{1}$ is the circle touching all the sides of the square $A B C D$ and $C_{2}$ is the circumcircle of square $A B C D$ . $L$ is a fixed line in the same plane. [2006]

Q. A line $L'$ through $A$ is drawn parallel to $B D$ . Point $S$ moves such that its distances from the line $B D$ and the vertex $A$ are equal. If locus of $S$ cuts $L'$ at $T_{2}$ and $T_{3}$ and $A C$ at $T_{1}$ , then area of $∆ T_{1} T_{2} T_{3}$ is

Smart Achievers · Accepted Answer

(3)

Since $S$ is equidistant from $A$ and line $B D$ , it traces a parabola.

Clearly, $A C$ is the axis, $A (1, 1)$ is the focus and $T_{1} (\frac{1}{2}, \frac{1}{2})$ is the vertex of the parabola.

$A T_{1} = \frac{1}{\sqrt{2}}$

$T_{2} T_{3} = latus rectum of parabola = 4 \times \frac{1}{\sqrt{2}} = 2 \sqrt{2}$

$∴$ $Area (∆ T_{1} T_{2} T_{3}) = \frac{1}{2} \times \frac{1}{\sqrt{2}} \times 2 \sqrt{2} = 1 sq. units$

Solution

1 $\frac{1}{2}$ sq. units

2 $\frac{2}{3}$ sq. units

3 $1$ sq. units

4 $2$ sq. units

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Solution

1 12 sq. units

2 23 sq. units

3 1 sq. units

4 2 sq. units

Ans. (3) Since S is equidistant from A and line BD, it traces a parabola. Clearly, AC is the axis, A(1,1) is the focus and T1(12,12) is the vertex of the parabola. AT1=12 T2T3=latus rectum of parabola=4×12=22 ∴ Area (∆T1T2T3)=12×12×22=1 sq. units

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1 $\frac{1}{2}$ sq. units

2 $\frac{2}{3}$ sq. units

3 $1$ sq. units

4 $2$ sq. units