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Yukti Vault

Coordinate Geometry

Points and Straight Lines: JEE Main & Advanced PYQs

284 previous year questions on Points and Straight Lines from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 51-100 of 284

  1. The distance between the two points $A$ and $A'$ which lie on $y=2$ such that both the line segments $AB$ and $A'B$ (where $B$ is the point $(2,3)$) subtend angle …
  2. If the line $l_1:3y-2x=3$ is the angular bisector of the lines $l_2:x-y+1=0$ and $l_3:\alpha x+\beta y+17=0$, then $\alpha^{2}+\beta^{2}-\alpha-\beta$ is equal to …
  3. The distance of the origin from the centroid of the triangle whose two sides have the equations $x-2y+1=0$ and $2x-y-1=0$ and whose orthocenter is …
  4. A ray of light passing through the point $P(2,3)$ reflects on the $x$-axis at point $A$ and the reflected ray passes through the point $Q(5,4)$. Let $R$ be the point …
  5. Let $ABC$ be an isosceles triangle in which $A$ is at $(-1,0)$, $\angle A=\frac{2\pi}{3}$, $AB=AC$ and $B$ is on the positive $x$-axis. If $BC=4\sqrt{3}$ and the line …
  6. The set of all possible values of $\theta$ in the interval $(0,\pi)$ for which the points $(1,2)$ and $(\sin\theta,\cos\theta)$ lie on the same side of the line $x+y=1$ …
  7. Let $A(-2,-1)$, $B(1,0)$, $C(\alpha,\beta)$ and $D(\gamma,\delta)$ be the vertices of a parallelogram $ABCD$. If the point $C$ lies on $2x-y=5$ and the point $D$ lies on …
  8. Two sides of a parallelogram are along the lines $4x+5y=0$ and $7x+2y=0$. If the equation of one of the diagonals of the parallelogram is $11x+7y=9$, then other diagonal …
  9. From the point $(-1,-1)$, two rays are sent making angles of $45^{\circ}$ with the line $x+y=0$. These rays get reflected from the mirror $x+2y=1$. If the equations of …
  10. Let $A(1,1)$, $B(-4,3)$, $C(-2,-5)$ be vertices of a triangle $ABC$, $P$ be a point on side $BC$, and $\Delta_1$ and $\Delta_2$ be the areas of triangles $APB$ and …
  11. Let $L$ denote the line in the $xy$-plane with $x$ and $y$ intercepts as $3$ and $1$ respectively. Then the image of the point $(-1,-4)$ in this line is :
  12. If the orthocentre of the triangle formed by the lines $2x+3y-1=0$, $x+2y-1=0$ and $ax+by-1=0$, is the centroid of another triangle, whose circumcentre and orthocentre …
  13. Let $A(0,a+2)$, $B(0,a)$, $C(-2,0)$ and $D(2,0)$ be four points and the lines $AD$ and $BC$ intersect at $P(x,y)$. If the locus of $P$ is the curve $f(x,y)=0$ and the …
  14. The point $P(a,b)$ undergoes the following three transformations successively : (a) reflection about the line $y=x$. (b) translation through $2$ units along the positive …
  15. The equations of the sides $AB$, $BC$ and $CA$ of a triangle $ABC$ are $2x+y=0$, $x+py=39$ and $x-y=3$ respectively and $P(2,3)$ is its circumcentre. Then which of the …
  16. Let a triangle be bounded by the lines $L_1:2x+5y=10$; $L_2:-4x+3y=12$ and the line $L_3$, which passes through the point $P(2,3)$, intersects $L_2$ at $A$ and $L_1$ at …
  17. If the perpendicular bisector of the line segment joining the points $P(1,4)$ and $Q(k,3)$ has $y$-intercept equal to $-4$, then a value of $k$ is :
  18. Let $\alpha_1,\alpha_2$ ($\alpha_1<\alpha_2$) be the values of $\alpha$ for the points $(\alpha,-3)$, $(2,0)$ and $(1,\alpha)$ to be collinear. Then the equation of the …
  19. Let the point $P(\alpha,\beta)$ be at a unit distance from each of the two lines $L_1:3x-4y+12=0$, and $L_2:8x+6y+11=0$. If $P$ lies below $L_1$ and above $L_2$, then …
  20. Let the circumcentre of a triangle with vertices $A(a,3)$, $B(b,5)$ and $C(a,b)$, $ab>0$ be $P(1,1)$. If the line $AP$ intersects the line $BC$ at the point …
  21. Let $C$ be the centroid of the triangle with vertices $(3,-1)$, $(1,3)$ and $(2,4)$. Let $P$ be the point of intersection of the lines $x+3y-1=0$ and $3x-y+1=0$. Then …
  22. If a $\Delta ABC$ has vertices $A(-1,7)$, $B(-7,1)$ and $C(5,-5)$, then its orthocentre has coordinates :
  23. The equations of the sides $AB$, $BC$ and $CA$ of a triangle $ABC$ are $2x+y=0$, $x+py=15a$ and $x-y=3$ respectively. If its orthocentre is $(2,a)$, $-\frac12<a<2$, then …
  24. If the area of $\triangle ABC$ with vertices $A(0,1)$, $B(2,3)$ and $C(4,b)$ is $6$ sq. units, then the positive value of '$b$' is $\_\_\_\_\_\_$ .
  25. Let $A,B$ be points on the two half-lines $x-\sqrt{3}|y|=\alpha$, $\alpha>0$ at a distance of $\alpha$ from their point of intersection $P$. The line segment $AB$ meets …
  26. Let $A(\alpha,-2)$, $B(\alpha,6)$ and $C\left(\frac\alpha4,-2\right)$ be vertices of a $\Delta ABC$. If $\left(5,\frac\alpha4\right)$ is the circumcentre of …
  27. A point $P$ moves on the line $2x-3y+4=0$. If $Q(1,4)$ and $R(3,-2)$ are fixed points, then the locus of the centroid of $\Delta PQR$ is a line :
  28. If the line $3x+4y-24=0$ intersects the $x$-axis at the point $A$ and the $y$-axis at the point $B$, then the incentre of the triangle $OAB$, where $O$ is the origin, is …
  29. Two sides of a parallelogram are along the lines $x+y=3$ and $x-y+3=0$. If its diagonals intersect at $(2,4)$, then one of its vertex is :
  30. If a tangent to the circle $x^{2}+y^{2}=1$ intersects the coordinate axes at distinct points $P$ and $Q$, then the locus of the mid-point of $PQ$ is :
  31. Two vertices of a triangle are $(0,2)$ and $(4,3)$. If its orthocentre is at the origin, then its third vertex lies in which quadrant?
  32. The points $\left(0,\frac83\right)$, $(1,3)$ and $(82,30)$
  33. A straight line through a fixed point $(2,3)$ intersects the coordinate axes at distinct points $P$ and $Q$. If $O$ is the origin and the rectangle $OPRQ$ is completed …
  34. Let the orthocentre and centroid of a triangle be $A(-3,5)$ and $B(3,3)$ respectively. If $C$ is the circumcentre of this triangle, then the radius of the circle having …
  35. Let $L$ be the line passing through the point $P(1,2)$ such that its intercepted segment between the co-ordinate axes is bisected at $P$. If $L_1$ is the line …
  36. Slope of a line passing through $P(2,3)$ and intersecting the line $x+y=7$ at a distance of $4$ units from $P$, is
  37. The locus of the mid-points of the perpendiculars drawn from points on the line $x=2y$, to the line $x=y$, is :
  38. Let $R$ be the interior region between the lines $3x-y+1=0$ and $x+2y-5=0$ containing the origin. The set of all values of $a$, for which the points …
  39. Two sides of a rhombus are along the lines $x-y+1=0$ and $7x-y-5=0$. If its diagonals intersect at $(-1,-2)$, then which one of the following is a vertex of this rhombus?
  40. The point $(2,1)$ is translated parallel to the line $L: x-y=4$ by $2\sqrt3$ units. If the new point $Q$ lies in the third quadrant, then the equation of the line …
  41. The distance of the point $(2,3)$ from the line $2x-3y+28=0$, measured parallel to the line $\sqrt3x-y+1=0$, is equal to
  42. Let the equations of two sides of a triangle be $3x-2y+6=0$ and $4x+5y-20=0$. If the orthocenter of this triangle is at $(1,1)$ then the equation of its third side is :
  43. The combined equation of the two lines $ax+by+c=0$ and $a'x+b'y+c'=0$ can be written as $(ax+by+c)(a'x+b'y+c')=0$. The equation of the angle bisectors of the lines …
  44. Let $B$ and $C$ be the two points on the line $y+x=0$ such that $B$ and $C$ are symmetric with respect to the origin. Suppose $A$ is a point on $y-2x=2$ such that …
  45. If a variable line drawn through the intersection of the lines $\frac x3+\frac y4=1$ and $\frac x4+\frac y3=1$, meets the coordinate axes at $A$ and $B$ ($A\neq B$) …
  46. Let $S$ be the set of all triangles in the $xy$-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. …
  47. Let $A$ be the point of intersection of the lines $3x+2y=14$, $5x-y=6$ and $B$ be the point of intersection of the lines $4x+3y=8$, $6x+y=5$. The distance of the point …
  48. If the straight line $2x-3y+17=0$ is perpendicular to the line passing through the points $(7,17)$ and $(15,\beta)$, then $\beta$ equals :
  49. If the orthocentre of the triangle, whose vertices are $(1,2)$, $(2,3)$ and $(3,1)$ is $(\alpha,\beta)$, then the quadratic equation whose roots are $\alpha+4\beta$ and …
  50. In a triangle $ABC$, coordinates of $A$ are $(1,2)$ and the equations of the medians through $B$ and $C$ are $x+y=5$ and $x=4$ respectively. Then area of $\triangle ABC$ …

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