Question

The joint equation of a pair of lines passing through point \((1,4)\), one of which is parallel to X-axis and the other makes an angle of \(45^{\circ}\) with the positive direction of X-axis, is

• A. \(x^2-xy-x+4y-12 = 0\)
• B. \(xy-y^2-4x+7y-12 = 0\)
• C. \(x^2+2xy-y^2+7 = 0\)
• D. \(xy-2y^2+3x+2y+17 = 0\)

Hint:

One line has slope 0, the other slope 1; multiply the two line equations.

Answer 1

The Correct Option is B

Step 1: Write each line
$y=4$ and $y=x+3$.

Step 2: Product of factors
$(y-4)(y-x-3)=y^2-xy-7y+4x+12$. Multiplying by $-1$ gives $xy-y^2-4x+7y-12=0$, option (B).

Final Answer:
The joint equation matches option (B). \[ \boxed{xy-y^2-4x+7y-12=0} \]