Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
AP EAMCET202317 May 2023Evening ShiftMathematicsStraight LinesActual

The lines L₁: y-x=0 and L₂: 2 x+y=0 intersect the line L₃: y+2=0 at P and Q respectively. The bisector of the acute angle between L ₁ and L ₂ intersects L ₃ at R . Statement 1: PR : RQ =2 2 : 5 Statement 2: In any triangle, the bisector of an angle divides the triangle into two similar triangles.

Options

  1. AStatement-1 is true, Statement-2 is false
  2. BStatement-1 is false, Statement- 2 is true
  3. CStatement-1 and Statement-2 are both true
  4. DStatement-1 and Statement- 2 are both false

Correct answer

A. Statement-1 is true, Statement-2 is false

Step-by-step solution

Given: L ₁: y-x=0 aligned & L ₂: 2 x+y=0 & ~L ₃=y+2=0 aligned Solving L ₁ and L ₃ , we get x=-2, y=-2 p(x, y)=(-2,-2) Solving L ₂: 2 x+y=0 & ~L ₃: y+2=0 , we get x=1 & y=-2 Q(x, y)=(1,-2) Now, OP = (0-(-2))^2+(0-(-2))^2 =2 2 OR = (0-1)^2+(0-(-2))^2 = 5 OR is the bisector of POQ , therefore R divided in the same ratio as O P : OQ. Thus, PR : RQ = AP : AQ =2 2 : 5 Statement-1 is true. Statement-2 is clearly false.

Practice Straight Lines on Quantrex Academy →

More from Straight Lines

The locus of the third vertex of a right-angled triangle, the ends of whose hypotenuse are (1,2) and (4,5) , is 2025The coordinate axes are rotated about the origin in the counter clockwise direction through an angle 60^ . If a and b are the intercepts made on the new axes by a straight line who 2025The image of a point (2,-1) with respect to the line x-y+1=0 is 2025If a straight line is at a distance of 10 units from the origin and the perpendicular drawn from the origin to it makes an angle 4 with the negative X -axis in the negative directi 2025A straight line passing through the origin O meets the parallel lines 4 x+2 y=9 and 2 x + y +6=0 at the points P and Q respectively. Then the point O divides the line segment PQ in 2025A (4,3), B (2,5) are two points. If P is a variable point on the same side as that of the origin with respect to the line AB and is at most at a distance of 5 units from the midpoi 2025By shifting the origin to the point (2,3) through translation of axes, if the equation of the curve x^2+3 x y-2 y^2+4 x-y-20=0 is transformed to the form Ax ^2+ Bxy + Cy ^2+ Dx + E 2025The points (2,3) and (-4,- 4 3 ) lie on the opposite sides of the line L 5 x -6 y + k =0 and k is an integer. If the points (1,2) and (4,5) lie on the same side of the line L =0 , 2025 Full Straight Lines list All AP EAMCET PYQs