Straight Lines
Straight Line
Allen Star Batch
Grade 11
Question:
Consider a variable line 'L' which passes through the point of intersection 'P' of the lines $3x + 4y - 12 = 0$ and $x + 2y - 5 = 0$ meeting the coordinate axes at points $A$ and $B$:
then the locus of middle point of the segment $AB$ has the equation $3x + 4y = 4xy$
then the locus of the feet of the perpendicular from the origin on the variable line 'L' has the equation $2(x^2 + y^2) - 4x - 3y = 0$
Locus of the centroid of the variable triangle $OAB$ has the equation (where 'O' is the origin) $3x + 4y - 6xy = 0$
Locus of the centroid of the variable triangle $OAB$ has the equation (where 'O' is the origin) $3x + 4y + 6xy = 0$
Step-by-Step Solution
Key Concept: The locus of a variable point is found by eliminating the parameter from the geometric constraints and intersection conditions.
The intersection of two lines occurs at $(\frac{2h}{3}, \frac{2k}{3})$. The equation of line $AB$ through $A(2h, 0)$ and $B(0, 2k)$ is $\frac{x}{2h} + \frac{y}{2k} = 1$. Substituting the intersection point and simplifying yields $4k + 3h = 4hk$, which rearranges to $3x + 4y = 4xy$ relating the coordinates.
Correct Answer: 1,2,3