Question

A straight line passes through a point \(A(2,5)\) and makes an angle of \(45^\circ\) with the positive X-axis when measured in positive direction. If this line intersects the line passing through the points \((1,-2)\) and \((3,-4)\) at \(B\), then \(AB=\)

• A. \(2\sqrt2\)
• B. \(5\sqrt2\)
• C. \(4\sqrt2\)
• D. \(\sqrt{82}\)

Hint:

In coordinate geometry problems involving intersection, always first write equations of both lines clearly. Solving simultaneous equations accurately is the key step before applying distance formula.

Answer 1

The Correct Option is B

Concept: To solve intersection of straight lines problems, the standard approach is:
• Determine equation of first line using point and slope.
• Determine equation of second line using two-point form.
• Solve both equations simultaneously to obtain intersection point.
• Use distance formula to determine required length. Slope of line making angle \(\theta\) with positive x-axis is \[ m=\tan\theta \] Distance formula between two points is \[ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \]

Step 1: Find equation of first line passing through A.
The line makes angle \[ 45^\circ \] Thus slope becomes \[ m=\tan45^\circ \] \[ m=1 \] Since line passes through point \[ (2,5) \] Using point slope form \[ y-y_1=m(x-x_1) \] Substitute values \[ y-5=1(x-2) \] \[ y=x+3 \] This is first line.

Step 2: Find equation of second line using two given points.
The second line passes through \[ (1,-2) \] and \[ (3,-4) \] Slope formula is \[ m=\frac{y_2-y_1}{x_2-x_1} \] Substituting values \[ m=\frac{-4-(-2)}{3-1} \] \[ =\frac{-2}{2} \] \[ =-1 \] Using point slope form \[ y+2=-1(x-1) \] \[ y+2=-x+1 \] \[ y=-x-1 \] This is second line.

Step 3: Find point of intersection B.
The two equations are \[ y=x+3 \] and \[ y=-x-1 \] Equating both \[ x+3=-x-1 \] \[ 2x=-4 \] \[ x=-2 \] Substituting into first equation \[ y=-2+3 \] \[ y=1 \] Thus point B is \[ (-2,1) \]

Step 4: Find distance AB.
Coordinates are \[ A=(2,5) \] \[ B=(-2,1) \] Distance formula gives \[ AB=\sqrt{(-2-2)^2+(1-5)^2} \] \[ =\sqrt{(-4)^2+(-4)^2} \] \[ =\sqrt{16+16} \] \[ =\sqrt{32} \] \[ =4\sqrt2 \] Thus required distance is \[ \boxed{4\sqrt2} \] Hence correct option becomes \[ \boxed{(3)} \]