Question

The point with integral coordinates on the line \(x+y=1\), that lie at a distance \(2\) units from the line \(5x+12y=0\), is

• A. \((-7,8)\)
• B. \((-1,2)\)
• C. \((-12,13)\)
• D. \((-2,3)\)
• E. \((-3,4)\)

Hint:

For questions involving a point on one line and fixed distance from another line, first use the line equation to reduce variables, then apply the point-to-line distance formula.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We need to find a point that satisfies two conditions: it lies on a specific line, and it is at a specific distance from another line. We can represent any point on the first line using a single parameter and then use the formula for the distance from a point to a line to solve for that parameter.
Step 2: Key Formula or Approach:
1. Represent a general point on the line \(x+y=1\). If we let the x-coordinate be \(h\), then from the equation, the y-coordinate must be \(1-h\). So, the point is \(P(h, 1-h)\).
2. The formula for the perpendicular distance (\(d\)) from a point \((x_1, y_1)\) to a line \(Ax + By + C = 0\) is:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \] Step 3: Detailed Explanation:
Let the required point be \(P(h, k)\). Since P lies on the line \(x+y=1\), we have \(k=1-h\). So, the coordinates of P are \((h, 1-h)\).
The second condition is that the distance from P to the line \(5x + 12y = 0\) is 2 units.
Using the distance formula with \((x_1, y_1) = (h, 1-h)\), \(A=5, B=12, C=0\), and \(d=2\):
\[ 2 = \frac{|5(h) + 12(1-h)|}{\sqrt{5^2 + 12^2}} \] \[ 2 = \frac{|5h + 12 - 12h|}{\sqrt{25 + 144}} \] \[ 2 = \frac{|12 - 7h|}{\sqrt{169}} \] \[ 2 = \frac{|12 - 7h|}{13} \] Multiply both sides by 13:
\[ 26 = |12 - 7h| \] This absolute value equation gives two possibilities:
Case 1: \(12 - 7h = 26\)
\[ -7h = 14 \] \[ h = -2 \] If \(h=-2\), then \(k = 1 - (-2) = 3\). The point is \((-2, 3)\). Since both coordinates are integers, this is a valid solution.
Case 2: \(12 - 7h = -26\)
\[ -7h = -38 \] \[ h = \frac{38}{7} \] This does not give an integral coordinate, so we discard this solution.
Step 4: Final Answer:
The only point with integral coordinates that satisfies the conditions is (-2, 3). Therefore, option (D) is correct.