Straight Lines
Straight Line
Allen Star Batch
Grade 11
Question:
For all values of $\theta$, the lines represented by the equation $(2\cos\theta + 3\sin\theta)x + (3\cos\theta - 5\sin\theta)y - (5\cos\theta - 2\sin\theta) = 0$
pass through a fixed point
Pass through the point $(1,1)$
pass through a fixed point whose reflection in the line $x + y = \sqrt{2}$ is $\left(\sqrt{2} - 1, \sqrt{2} - 1\right)$
pass through the origin
Step-by-Step Solution
Key Concept: An equation of the form $L_1 + \lambda L_2 = 0$ represents all lines passing through the intersection of $L_1$ and $L_2$.
The equation $(2x+3y-5)\cos\theta + (3x-5y+2)\sin\theta = 0$ can be rewritten as $(2x+3y-5) + \tan\theta(3x-5y+2) = 0$, which represents $L_4 + \lambda L_2 = 0$. This is a family of lines passing through the intersection point of $L_4: 2x+3y-5=0$ and $L_2: 3x-5y+2=0$, which is $(1,1)$. The mirror image of $(1,1)$ about the line $y=x$ is $(\sqrt{2}-1, \sqrt{2}-1)$.
Correct Answer: 1,2,3