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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 251-284 of 284

  1. The coordinates of $A,B,C$ are $(6,3)$, $(-3,5)$, $(4,-2)$ respectively, and $P$ is any point $(x,y)$. Show that the ratio of the area of the triangles $\triangle PBC$ …
  2. Let a given line $L_1$ intersect the $x$ and $y$ axes at $P$ and $Q$, respectively. Let another line $L_2$, perpendicular to $L_1$, cut the $x$ and $y$ axes at $R$ and …
  3. Let $ABC$ be a triangle with $AB=AC$. If $D$ is the midpoint of $BC$, $E$ the foot of the perpendicular drawn from $D$ to $AC$ and $F$ the mid-point of $DE$, prove that …
  4. Using co-ordinate geometry, prove that the three altitudes of any triangle are concurrent.
  5. Let $S$ be a square of unit area. Consider any quadrilateral which has one vertex on each side of $S$. If $a$, $b$, $c$ and $d$ denote the lengths of the sides of the …
  6. A straight line $L$ through the origin meets the lines $x+y=1$ and $x+y=3$ at $P$ and $Q$ respectively. Through $P$ and $Q$ two straight lines $L_1$ and $L_2$ are drawn …
  7. A line is drawn through the point $(1,2)$ to meet the coordinate axes at $P$ and $Q$ such that it forms a triangle $OPQ$, where $O$ is the origin. If the area of the …
  8. If the line $2x+y=k$ passes through the point which divides the line segment joining the points $(1,1)$ and $(2,4)$ in the ratio $3:2$, then $k$ equals :
  9. From a point $O$ inside a triangle $ABC$, perpendiculars $OD$, $OE$, $OF$ are drawn to the sides $BC$, $CA$, $AB$ respectively. Prove that the perpendiculars from $A$ …
  10. One side of a square lies along the line $x-3y+1=0$ and its one vertex is $(1,4)$. If $(a,b)$ and $(c,d)$ are its two vertices on the line $x-3y+1=0$, then …
  11. Let a vector $\alpha\hat{i}+\beta\hat{j}$ be obtained by rotating the vector $\sqrt{3}\hat{i}+\hat{j}$ by an angle $45^{\circ}$ about the origin in counterclockwise …
  12. The vertices of a triangle are $A(-1,3)$, $B(-2,2)$ and $C(3,-1)$. A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation …
  13. Consider a triangle $ABC$ having the vertices $A(1,2)$, $B(\alpha,\beta)$ and $C(\gamma,\delta)$ and angles $\angle ABC=\dfrac{\pi}{6}$ and $\angle BAC=\dfrac{2\pi}{3}$. …
  14. Let two straight lines drawn from the origin $O$ intersect the line $3x+4y=12$ at the points $P$ and $Q$ such that $\triangle OPQ$ is an isosceles triangle and …
  15. Let $A(-1,1)$ and $B(2,3)$ be two points and $P$ be a variable point above the line $AB$ such that the area of $\triangle PAB$ is $10$. If the locus of $P$ is …
  16. A ray of light coming from the point $\left(2,2\sqrt{3}\right)$ is incident at an angle $30^{\circ}$ on the line $x=1$ at the point $A$. The ray gets reflected on the …
  17. Let the triangle $PQR$ be the image of the triangle with vertices $(1,3)$, $(3,1)$ and $(2,4)$ in the line $x+2y=2$. If the centroid of $\triangle PQR$ is the point …
  18. Let the distance between two parallel lines be $5$ units and a point $P$ lie between the lines at a unit distance from one of them. An equilateral triangle $PQR$ is …
  19. Let $A(6,8)$, $B(10\cos\alpha,-10\sin\alpha)$ and $C(-10\sin\alpha,10\cos\alpha)$, be the vertices of a triangle. If $L(a,9)$ and $G(h,k)$ be its orthocenter and …
  20. Let the area of a $\triangle PQR$ with vertices $P(5,4)$, $Q(-2,4)$ and $R(a,b)$ be $35$ square units. If its orthocenter and centroid are …
  21. Let the points $\left(\dfrac{11}{2},\alpha\right)$ lie on or inside the triangle with sides $x+y=11$, $x+2y=16$ and $2x+3y=29$. Then the product of the smallest and the …
  22. Two equal sides of an isosceles triangle are along $-x+2y=4$ and $x+y=4$. If $m$ is the slope of its third side, then the sum, of all possible distinct values of $m$, is:
  23. The pair of lines represented by $3ax^{2}+5xy+\left(a^{2}-2\right)y^{2}=0$ are perpendicular to each other for :
  24. If the pair of lines $ax^{2}+2hxy+by^{2}+2gx+2fy+c=0$ intersect on the $y$-axis then :
  25. If the pair of straight lines $x^{2}-2pxy-y^{2}=0$ and $x^{2}-2qxy-y^{2}=0$ be such that each pair bisects the angle between the other pair, then :
  26. If one of the lines given by $6x^{2}-xy+4cy^{2}=0$ is $3x+4y=0$, then $c$ equals :
  27. If the sum of the slopes of the lines given by $x^{2}-2cxy-7y^{2}=0$ is four times their product, then $c$ has the value :
  28. Let $P=(-1,0)$, $Q=(0,0)$ and $R=\left(3,3\sqrt3\right)$ be three points. The equation of the bisector of the angle $PQR$ is :
  29. If one of the lines of $my^{2}+\left(1-m^{2}\right)xy-mx^{2}=0$ is a bisector of the angle between the lines $xy=0$, then $m$ is :
  30. The lines $L_1:y-x=0$ and $L_2:2x+y=0$ intersect the line $L_3:y+2=0$ at $P$ and $Q$ respectively. The bisector of the acute angle between $L_1$ and $L_2$ intersects …
  31. Let $PQR$ be a triangle. The points $A$, $B$ and $C$ are on the sides $QR$, $RP$ and $PQ$ respectively such that $\frac{QA}{AR}=\frac{RB}{BP}=\frac{PC}{CQ}=\frac12$. …
  32. Let the line $x+y=1$ meet the axes of $x$ and $y$ at $A$ and $B$, respectively. A right angled triangle $AMN$ is inscribed in the triangle $OAB$, where $O$ is the origin …
  33. A man starts walking from the point $P(-3,4)$, touches the $x$-axis at $R$, and then turns to reach at the point $Q(0,2)$. The man is walking at a constant speed. If the …
  34. Consider a triangle having vertices $A(-2,3)$, $B(1,9)$ and $C(3,8)$. If a line $L$ passing through the circum-centre of triangle $ABC$, bisects line $BC$, and …

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