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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 1-50 of 284

  1. The intersection of three lines $x-y=0$, $x+2y=3$ and $2x+y=6$ is a :
  2. A line passing through the point $A(9,0)$ makes an angle of $30^{\circ}$ with the positive direction of $x$-axis. If this line is rotated about $A$ through an angle of …
  3. Let $R$ be the point $(3,7)$ and let $P$ and $Q$ be two points on the line $x+y=5$ such that $PQR$ is an equilateral triangle. Then the area of $\triangle PQR$ is :
  4. Let the three sides of a triangle are on the lines $4x-7y+10=0$, $x+y=5$ and $7x+4y=15$. Then the distance of its orthocentre from the orthocentre of the triangle formed …
  5. Let the centroid of an equilateral triangle $ABC$ be at the origin. Let one of the sides of the equilateral triangle be along the straight line $x+y=3$. If $R$ and $r$ …
  6. Let $A$ be a fixed point $(0,6)$ and $B$ be a moving point $(2t,0)$. Let $M$ be the mid-point of $AB$ and the perpendicular bisector of $AB$ meets the $y$-axis at $C$. …
  7. Let $A(-1,1)$, $B(3,4)$ and $C(2,0)$ be given three points. A line $y=mx$, $m>0$, intersects lines $AC$ and $BC$ at point $P$ and $Q$ respectively. Let $A_1$ and $A_2$ …
  8. If $x^{2}-y^{2}+2hxy+2gx+2fy+c=0$ is the locus of a point, which moves such that it is always equidistant from the lines $x+2y+7=0$ and $2x-y+8=0$, then the value of …
  9. The portion of the line $4x+5y=20$ in the first quadrant is trisected by the lines $L_1$ and $L_2$ passing through the origin. The tangent of an angle between the lines …
  10. If the line segment joining the points $(5,2)$ and $(2,a)$ subtends an angle $\frac{\pi}{4}$ at the origin, then the absolute value of the product of all possible values …
  11. Let the equation of the pair of lines, $y=px$ and $y=qx$, can be written as $(y-px)(y-qx)=0$. Then the equation of the pair of the angle bisectors of the lines …
  12. Let $A(a,0)$, $B(b,2b+1)$ and $C(0,b)$, $b\neq0$, $|b|\neq1$, be points such that the area of triangle $ABC$ is $1$ sq. unit, then the sum of all possible values of $a$ …
  13. If $p$ and $q$ are the lengths of the perpendiculars from the origin on the lines, $x\operatorname{cosec}\alpha-y\sec\alpha=k\cot2\alpha$ and …
  14. If $(k,3)$ is an interior point of the triangle formed by the lines, $x+y=4$, $x-y=5$ and $x-2y+2=0$, then the set of all values of $k$ is the interval
  15. In a $\triangle ABC$, suppose $y=x$ is the equation of the bisector of the angle $B$ and the equation of the side $AC$ is $2x-y=2$. If $2AB=BC$ and the points $A$ and …
  16. If $P(6,1)$ be the orthocentre of the triangle whose vertices are $A(5,-2)$, $B(8,3)$ and $C(h,k)$, then the point $C$ lies on the circle :
  17. A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $L_1:2x+y+6=0$ and $L_2:4x+2y-p=0$, $p>0$, at the …
  18. If $(\alpha,\beta)$ is the orthocenter of the triangle $ABC$ with vertices $A(3,-7)$, $B(-1,2)$ and $C(4,5)$, then $9\alpha-6\beta+60$ is equal to
  19. Let $\alpha,\beta,\gamma,\delta\in\mathbb{Z}$ and let $A(\alpha,\beta)$, $B(1,0)$, $C(\gamma,\delta)$ and $D(1,2)$ be the vertices of a parallelogram $ABCD$. If …
  20. Let $C(\alpha,\beta)$ be the circumcenter of the triangle formed by the lines $4x+3y=69$, $4y-3x=17$, and $x+7y=61$. Then $(\alpha-\beta)^{2}+\alpha+\beta$ is equal to
  21. Let $A(a,b)$, $B(3,4)$ and $C(-6,-8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $P(2a+3,7b+5)$ from …
  22. In a triangle $PQR$, the co-ordinates of the points $P$ and $Q$ are $(-2,4)$ and $(4,-2)$ respectively. If the equation of the perpendicular bisector of $PR$ is …
  23. In an isosceles triangle $ABC$, the vertex $A$ is $(6,1)$ and the equation of the base $BC$ is $2x+y=4$. Let the point $B$ lie on the line $x+3y=7$. If $(\alpha,\beta)$ …
  24. A ray of light coming from the point $P(1,2)$ gets reflected from the point $Q$ on the $x$-axis and then passes through the point $R(4,3)$. If the point $S(h,k)$ is such …
  25. If the orthocenter of the triangle formed by the lines $y=x+1$, $y=4x-8$ and $y=mx+c$ is at $(3,-1)$, then $m-c$ is :
  26. Two vertices of a triangle $ABC$ are $A(3,-1)$ and $B(-2,3)$, and its orthocentre is $P(1,1)$. If the coordinates of the point $C$ are $(\alpha,\beta)$ and the centre of …
  27. Let $a$ be the length of a side of a square $OABC$ with $O$ being the origin. Its side $OA$ makes an acute angle $\alpha$ with the positive $x$-axis and the equations of …
  28. The equations of two sides $AB$ and $AC$ of a triangle $ABC$ are $4x+y=14$ and $3x-2y=5$, respectively. The point $\left(2,-\frac{4}{3}\right)$ divides the third side …
  29. Let $ABC$ be the triangle such that the equations of lines $AB$ and $AC$ be $3y-x=2$ and $x+y=2$, respectively, and the points $B$ and $C$ lie on $x$-axis. If $P$ is the …
  30. Let $R$ be a rectangle given by the lines $x=0$, $x=2$, $y=0$ and $y=5$. Let $A(\alpha,0)$ and $B(0,\beta)$, $\alpha\in[0,2]$ and $\beta\in[0,5]$, be such that the line …
  31. Let $ABC$ be a triangle with $A(-3,1)$ and $\angle ACB=\theta$, $0<\theta<\frac{\pi}{2}$. If the equation of the median through $B$ is $2x+y-3=0$ and the equation of …
  32. Let $(\alpha,\beta)$ be the centroid of the triangle formed by the lines $15x-y=82$, $6x-5y=-4$ and $9x+4y=17$. Then $\alpha+2\beta$ and $2\alpha-\beta$ are the roots of …
  33. If the point $\left(\alpha,\frac{7\sqrt{3}}{3}\right)$ lies on the curve traced by the mid-points of the line segments of the lines $x\cos\theta+y\sin\theta=7$ …
  34. A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $\frac{1}{4}$. Three stones $A$, $B$ …
  35. If a straight line drawn through the point of intersection of the lines $4x+3y-1=0$ and $3x+4y-1=0$, meets the co-ordinate axes at the points $P$ and $Q$, then the locus …
  36. Let $A$ be the set of all points $(\alpha,\beta)$ such that the area of triangle formed by the points $(5,6)$, $(3,2)$ and $(\alpha,\beta)$ is $12$ square units. Then …
  37. The straight lines $l_1$ and $l_2$ pass through the origin and trisect the line segment of the line $L:9x+5y=45$ between the axes. If $m_1$ and $m_2$ are the slopes of …
  38. The image of the point $(3,5)$ in the line $x-y+1=0$, lies on :
  39. The equation of one of the straight lines which passes through the point $(1,3)$ and makes an angle $\tan^{-1}(\sqrt{2})$ with the straight line, $y+1=3\sqrt{2}\,x$ is :
  40. Consider the lines $x(3\lambda+1)+y(7\lambda+2)=17\lambda+5$, $\lambda$ being a parameter, all passing through a point $P$. One of these lines (say $L$) is farthest from …
  41. The number of integral values of $m$ so that the abscissa of point of intersection of lines $3x+4y=9$ and $y=mx+1$ is also an integer, is :
  42. Let the area of the triangle with vertices $A(1,\alpha)$, $B(\alpha,0)$ and $C(0,\alpha)$ be $4$ sq. units. If the points $(\alpha,-\alpha)$, $(-\alpha,\alpha)$ and …
  43. Let the area of the triangle formed by a straight line $L:x+by+c=0$ with co-ordinate axes be $48$ square units. If the perpendicular drawn from the origin to the line …
  44. A triangle $ABC$ lying in the first quadrant has two vertices as $A(1,2)$ and $B(3,1)$. If $\angle BAC=90^\circ$, and $\mathrm{ar}(\Delta ABC)=5\sqrt5$ sq. units, then …
  45. A line, with the slope greater than one, passes through the point $A(4,3)$ and intersects the line $x-y-2=0$ at the point $B$. If the length of the line segment $AB$ is …
  46. If the line, $2x-y+3=0$ is at a distance $\frac1{\sqrt5}$ and $\frac2{\sqrt5}$ from the lines $4x-2y+\alpha=0$ and $6x-3y+\beta=0$, respectively, then the sum of all …
  47. Let the equations of two adjacent sides of a parallelogram $ABCD$ be $2x-3y=-23$ and $5x+4y=23$. If the equation of its one diagonal $AC$ is $3x+7y=23$ and the distance …
  48. Let two points be $A(1,-1)$ and $B(0,2)$. If a point $P(x',y')$ be such that the area of $\Delta PAB=5$ sq. units and it lies on the line, $3x+y-4\lambda=0$, then a …
  49. Let a ray of light passing through the point $(3,10)$ reflects on the line $2x+y=6$ and the reflected ray passes through the point $(7,2)$. If the equation of the …
  50. Let $m_1,m_2$ be the slopes of two adjacent sides of a square of side $a$ such that $a^2+11a+3(m_1^2+m_2^2)=220$. If one vertex of the square is …

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