Question

Q is a variable point whose locus is 2x + 3y + 4 = 0; corresponding to a particular position of Q, P is the point of section of OQ, O being the origin, such that OP : PQ = 3 : 1. Find the locus of P?

• A. 2x + 3y + 3 = 0
• B. 2x - 3y + 3 = 0
• C. 3x + 2y + 3 = 0
• D. 3x - 2y + 3 = 0

Answer 1

Answer: (A)

2x + 3y + 3 = 0

Answer 2

Let P be $(h, k)$ and Q be $(x_Q, y_Q)$. O is the origin $(0,0)$. Since P divides OQ in the ratio OP : PQ = 3 : 1, we use the section formula: $h = \frac{1 \cdot x_O + 3 \cdot x_Q}{1+3} = \frac{0 + 3x_Q}{4} \implies x_Q = \frac{4h}{3}$ $k = \frac{1 \cdot y_O + 3 \cdot y_Q}{1+3} = \frac{0 + 3y_Q}{4} \implies y_Q = \frac{4k}{3}$ Q lies on the line $2x + 3y + 4 = 0$. Substitute the expressions for $x_Q$ and $y_Q$: $2\left(\frac{4h}{3}\right) + 3\left(\frac{4k}{3}\right) + 4 = 0$ Simplify the equation: $\frac{8h}{3} + 4k + 4 = 0$ Multiply by 3: $8h + 12k + 12 = 0$ Divide by 4: $2h + 3k + 3 = 0$ The locus of P is $2x + 3y + 3 = 0$.