Let $$\mathrm{O(0,0), P(3,4), Q(6,0)}$$ be the vertices of the triangle OPQ. The point R inside the triangle OPQ is such that the triangles OPR, PQR, OQR are of equal area. The coordinates of R are
Lines $$\mathrm{L}_{1}: y-x=0$$ and $$\mathrm{L}_{2}: 2 x+y=0$$ intersect the line $$\mathrm{L}_{3}: y+2=0$$ at $$\mathrm{P}$$ and $$\mathrm{Q}$$, respectively. The bisector of the acute angle between $$L_{1}$$ and $$L_{2}$$ intersects $$L_{3}$$ at $$R$$.
STATEMENT - 1 : The ratio PR : RQ equals $$2 \sqrt{2}: \sqrt{5}$$.
STATEMENT - 2 : In any triangle, bisector of an angle divides the triangle into two similar triangles.
Consider the following linear equations
$$ax + by + cz = 0$$
$$bx + cy + az = 0$$
$$cx + ay + bz = 0$$
Match the conditions/expressions in Column I with statements in Column II.
| Column I | Column II | ||
|---|---|---|---|
| (A) | $$a + b + c \ne 0$$ and $${a^2} + {b^2} + {c^2} = ab + bc + ca$$ | (P) | the equations represent planes meeting only at a single point. |
| (B) | $$a + b + c = 0$$ and $${a^2} + {b^2} + {c^2} \ne ab + bc + ca$$ | (Q) | the equations represent the line $$x=y=z$$. |
| (C) | $$a + b + c \ne 0$$ and $${a^2} + {b^2} + {c^2} \ne ab + bc + ca$$ | (R) | the equations represent identical planes. |
| (D) | $$a + b + c = 0$$ and $${a^2} + {b^2} + {c^2} = ab + bc + ca$$ | (S) | the equations represent the whole of the three dimensional space. |
The area of the triangle formed by the intersection of a line parallel to X-axis and passing through $$(h, k)$$ with the lines $$y=x$$ and $$x+y=2$$ is $$4 h^{2}$$. Find the locus of point $$P$$.
JEE Advanced Subjects
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