Two Dimensional Geometry jee mains questions | JEE Main Two Dimensional Geometry Previous Year Question

Q 31 :

The line passing through the point of intersection of $x + y = 2$ , $x - y = 0$ and is parallel to $x + 2 y = 5$ is $a x + b y = c$ , then value of $a + b + c$ is _________.

(6)

$Required equation of line which is parallel to given line x + 2 y = 5 is x + 2 y + k = 0$ $\dots (i)$

$Given equations of lines are$

$x + y = 2$ $\dots (i i)$

$x - y = 0$ $\dots (i i i)$

$Adding (ii) and (iii), we get 2 x = 2$ $\Rightarrow x = 1$

$Substituting x = 1 in (iii), we get y = 1$

$∴$ $Point of intersection is (1, 1)$

$Substituting x = 1, y = 1 in (i), we get$

$1 + 2 + k = 0$ $\Rightarrow k = - 3$

$∴$ $Required equation of line is x + 2 y = 3$

$Here, a = 1, b = 2 and c = 3$

$∴$ $a + b + c = 1 + 2 + 3 = 6$

Q 32 :

$A_{1}, A_{2}, \dots, A_{n}$ are points on the line $y = x$ lying in the positive quadrant such that $O A_{n} = n \cdot O A_{n - 1},$ $O$ being the origin. If $O A_{1} = 1$ and the coordinates of $A_{n}$ are $(2520 \sqrt{2}, 2520 \sqrt{2})$ , then the value of $n$ is _______.

(7)

$We have, O A_{n} = n \cdot O A_{n - 1}$

$= n (n - 1) \cdot O A_{n - 2} = \dots = n! \cdot O A_{1} = n!$

$∴$ $O A_{n} = n!$

$\Rightarrow \sqrt{2} \times 2520 \sqrt{2} = n!$

$\Rightarrow n! = 5040$

$\Rightarrow n = 7$

Q 33 :

The area of the rhombus $A B C D$ is 24. The equation of the diagonal BD is $4 x + 3 y + 2 = 0$ and $A = (3, 2)$ . The length of the side of the rhombus is ______.

(5)

$Let A C and B D intersect at P .$

$Now, A P = \frac{12 + 6 + 2}{\sqrt{16 + 9}} = 4$

$∴$ $Area of ∆ A B D = A P \times B P = \frac{24}{2} = 12$ $\Rightarrow B P = 3$

$∴$ $A B = \sqrt{A P^{2} + B P^{2}} = 5$

Q 34 :

In $Δ A B C$ , the equation of altitudes AM and BN are $x + 5 y - 3 = 0$ , $x + y - k = 0$ . If the altitude CL is given by $3 x - y - 1 = 0$ , then $k =$ ______.

(1)

$Solving the altitudes A M and C L, we get$

$Orthocentre = (\frac{1}{2}, \frac{1}{2})$ $lies on x + y - k = 0$ $\Rightarrow k = 1$

Q 35 :

The centroid of the triangle formed by $(a, b), (b, c), (c, a)$ is the origin and $a^{3} + b^{3} + c^{3} = k a b c,$ then $k$ is _______.

(3)

$Given, G = (\frac{a + b + c}{3}, \frac{a + b + c}{3}) = (0, 0)$

$\Rightarrow a + b + c = 0$

$\Rightarrow a^{3} + b^{3} + c^{3} = 3 a b c$

$\Rightarrow k = 3$

Q 36 :

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:

$x^{2} + y^{2} - 61 = 0$
$x^{2} + y^{2} - 52 = 0$
$x^{2} + y^{2} - 65 = 0$
$x^{2} + y^{2} - 74 = 0$

1

2

3

4

(3)

$Slope of AD = 3,$ $Slope of BC = - \frac{1}{3},$ $Equation of BC = 3 y + x - 17 = 0$

$Slope of BE = 1,$ $Slope of AC = - 1,$ $Equation of AC is x + y - 3 = 0,$ $point C is (- 4, 7)$

Q 37 :

If a straight line having negative slope passing through the point P = (8, 2) meets the x-axis at A and y-axis at B respectively. Then the minimum value of OA + OB (O being origin) is _____

20
18
10
15

1

2

3

4

(2)

$Equation of line is y - 2 = m (x - 8),$ $O A = 8 + \frac{2}{- m}$ and $O B = 8 (- m) + 2$

$O A + O B = 10 + 8 (- m) + \frac{2}{- m} \geq 18$ $(AM \geq GM)$

Q 38 :

In a $∆ A B C, A B = A C$ where $A$ is (3, 1) and the equation of the base $B C$ is $2 x + y = 4$ . Also, $B$ lies on $x + 3 y = 7$ . Then sum of coordinates of vertex $' C'$ is ______

$\frac{13}{5}$
$\frac{18}{5}$
$\frac{16}{5}$
$\frac{7}{5}$

1

2

3

4

(4)

$B = (1, 2) & Let C \equiv (α, 4 - 2 α)$

$∴$ $A B = \sqrt{2^{2} + 1^{2}} = \sqrt{{(2 α - 3)}^{2} + {(3 - α)}^{2}},$

$5 α^{2} - 18 α + 13 = 0$

Q 39 :

Consider the following Statements

Statement I: General form $a x + b y + c = 0$ (here $a, b$ and $c$ are the parameters and $a, b, c \in R$ ) always represents a straight line.

Statement II: A line $a x + b y + 1 = 0$ , is such that the algebraic sum of the perpendiculars on it, from a number of points $(x_{i}, y_{i}),$ $(i = 1, 2, \dots, n)$ , is zero, then the line always passes through a fixed point $(\bar{x}, \bar{y})$ where $\bar{x} = \frac{\sum x_{i}}{n}$ and $\bar{y} = \frac{\sum y_{i}}{n}$

Only Statement I is true
Only Statement II is true
Both Statement I and II are True
Both Statement I and II are false

1

2

3

4

(2)

$S_{1} : a = b = c = 0 Then it is not linear$

$S_{2} is obviously True$

Q 40 :

The equations of the sides AB, BC and CA of a triangle ABC are:

$2 x + y = 0, x + p y = 21 a, (a \neq 0)$ and $x - y = 3$ respectively. Let $P (2, a)$ be the centroid of $∆ A B C$ . Then ${(B C)}^{2}$ is equal to

25
36
121
122

1

2

3

4

(4)

$2 x + y = 0 ------(1)$

$x - y = 0 ------(2)$

$x + p y = 21 a ------(3)$

$Solving (1) & (2) \Rightarrow A (1, - 2)$

$Centroid of triangle A B C is (\frac{4 + s + t}{3}, \frac{- 2 - 2 s + t}{3}) = (2, a)$

$\Rightarrow s + t = 2 ------(4)$

$\Rightarrow s = - a, t = 2 + a$

$- 2 s + t = 3 a + 2 ------(5)$

$Solving (4) & (5) we get$

$B (- a, 2 a), C (a + 5, a + 2)$

$∴$ $Distance {(B C)}^{2} = 122$

Q 41 :

Let the circumcenter of a triangle with vertices $A (a, 3), B (b, 5)$ and $C (a, b), a b > 0$ be P(1, 1). If the line AP intersects the line BC at the point $Q (k_{1}, k_{2})$ , then $k_{1} + k_{2}$ is equal to

$2$
$\frac{4}{7}$
$\frac{2}{7}$
$4$

1

2

3

4

(2)

$slope of A C = \infty$

$Slope of P D = 0$

$D (\frac{a + a}{2}, \frac{b + 3}{2}) = D (a, \frac{b + 3}{2})$

$\frac{b + 3}{2} - 1 = 0$

$b + 3 - 2 = 0$

$\Rightarrow b = - 1$

$b = - 1$

$E (\frac{b + a}{2}, \frac{5 + b}{2}) = (\frac{a - 1}{2}, 2)$

$slope of B C \times slope of E P = - 1$

$(\frac{5 - b}{b - a}) \times (\frac{2 - 1}{\frac{a - 1}{2} - 1}) = - 1$

$\Rightarrow (\frac{6}{- 1 - a}) \times (\frac{2}{a - 3}) = - 1$

$\Rightarrow 12 = (1 + a) (a - 3)$

$\Rightarrow 12 = a^{2} - 3 a + a - 3$ $\Rightarrow a^{2} - 2 a - 15 = 0$

$a = - 3 accept$

$Equation of A P, A (- 3, 3), P (1, 1)$

$y - 1 = (\frac{3 - 1}{- 3 - 1}) (x - 1)$ $\Rightarrow x + 2 y = 3$

$Equation of B C, B (- 1, 5), C (- 3, - 1)$

$y + 1 = (\frac{5 + 1}{- 1 + 3}) (x + 3)$ $= 3 x + 9$

$3 x - y + 8 = 0$

$Q \equiv (\frac{- 13}{7}, \frac{17}{7})$

Q 42 :

Consider the following statements:

S1: For the line $2 x + 3 y - 7 = 0$ , the points origin and (1, 1) are on the same side.

S2: Two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ lie on the same side of the line $a x + b y + c = 0$ $(a, b, c \in R - {0})$ if $(a x_{1} + b y_{1} + c) (a x_{2} + b y_{2} + c) > 0$

Both statements are true
Both statements are false
S1 is true and S2 is false
S1 is false and S2 is true

1

2

3

4

(1)

Substituting the points in the given line gives negative values for both points

Q 43 :

If A(-2, -1), B(-2, 1), C(2, 1) are the vertices of a triangle ABC, then the perpendicular distance from its circumcenter to the side BC is

1
2
3
4

1

2

3

4

(1)

$△ A B C$ is a right angled triangle at B and midpoint of hypotenuse AC is the circumcentre

Circumcentre of $△ A B C$ is S(0,0)

Q 44 :

Let the line $L_{1} : x + 3 = 0$ intersect the lines $L_{2} : x - y = 0$ and $L_{3} : 3 x + y = 0$ at the points A and B, respectively. Let the bisector of the obtuse angle between the lines $L_{2}$ and $L_{3}$ intersect the line $L_{1}$ at the point C. Then $B C^{2} : A C^{2}$ is equal to: [2026]

5 : 1
1 : 5
2 : 3
3 : 2

1

2

3

4

(1)

We have, A : (–3, –3) and B : (–3, 9)

The equations of the angle bisectors given by:

$\frac{x - y}{\sqrt{1^{2} + {(- 1)}^{2}}} = \pm \frac{3 x + y}{\sqrt{3^{2} + 1^{2}}}$

$\Rightarrow \frac{x - y}{\sqrt{2}} = \pm \frac{3 x + y}{\sqrt{10}}$

$\Rightarrow \sqrt{5} (x - y) = \pm (3 x + y)$

To find the obtuse angle bisector, we check the sign of $a_{1} a_{2} + b_{1} b_{2}$ for the lines $L_{2} (x - y = 0)$ and $L_{3} (3 x + y = 0)$ , where $a_{1} = 1$ , $b_{1} = - 1$ , $a_{2} = 3$ , $b_{2} = 1$ : $a_{1} a_{2} + b_{1} b_{2} = (1) (3) + (- 1) (1) = 3 - 1 = 2$

Since, the result is positive, the obtuse angle bisector is given by the equation with the minus sign (opposite the sign of $a_{1} a_{2} + b_{1} b_{2}$ ):

$\sqrt{5} (x - y) = - (3 x + y)$

$\Rightarrow \sqrt{5} x - \sqrt{5} y = - 3 x - y$

$\Rightarrow y (1 - \sqrt{5}) = - x (3 x + \sqrt{5})$

$\Rightarrow y = x \frac{3 + \sqrt{5}}{\sqrt{5} - 1}$

Point C lies on $L_{1} = (x = - 3)$ , so substituting x = –3 into the bisector equations:

$y_{C} = (- 3) \frac{3 + \sqrt{5}}{\sqrt{5} - 1} = - 3 \frac{(3 + \sqrt{5}) (\sqrt{5} + 1)}{(\sqrt{5} - 1) (\sqrt{5} + 1)}$

$\Rightarrow y_{C} = - 3 \frac{3 \sqrt{5} + 3 + 3 + \sqrt{5}}{5 - 1}$

$= - 3 \frac{8 + 4 \sqrt{5}}{4} = - 3 (2 + \sqrt{5})$

$C = (- 3, - 6 - 3 \sqrt{5})$

$∴ \frac{B C^{2}}{A C^{2}} = \frac{90 (3 + \sqrt{5})}{18 (3 + \sqrt{5})} = 5 : 1$

Q 45 :

Let the vertex A of a triangle ABC be (1, 2) and the midpoint of the side AB be(5, –1). If the centroid of this triangle is (3, 4) and its circumcenter is $(α, β)$ , then $21 (α + β)$ is equal to: [2026]

309
403
497
524

1

2

3

4

(3)

From the given information:

FIGURE

Coordinates of $B : \frac{x_{2} + 1}{2} = 5; \frac{y_{2} + 2}{2} = - 1$

$\Rightarrow x_{2} = 9, y_{2} = - 4$

$∴ B \equiv (9, - 4)$

Coordinates of $C : \frac{1 + 9 + x_{3}}{3} = 3$

$\Rightarrow x_{3} = - 1 and \frac{2 - 4 + y_{3}}{3} = 4$

$\Rightarrow y_{3} = 14$

$∴ C \equiv (- 1, 14)$

Slope of $⊥^{r}$ bisector of AB, $m_{1} = \frac{4}{3}$

Equation of $⊥^{r}$ bisector of AB

$y + 1 = \frac{4}{3} (x - 5)$

$\Rightarrow 4 x - 3 y + 23 = 0$ ... (i)

Midpoint of BC is N (4, 5)

Slope of $⊥^{r}$ bisector of BC, $m_{2} = \frac{5}{9}$

Equation of $⊥^{r}$ bisector of BC,

$y - 5 = \frac{5}{9} (x - 4) \Rightarrow 9 y - 45 = 5 x - 20$

$\Rightarrow 5 x - 9 y + 25 = 0$ ... (ii)

Solving (i) & (ii), we get

$x = \frac{94}{7}, y = \frac{4 \times 94}{21} - \frac{23}{3} = \frac{215}{21}$

$∴ (α, β) = (\frac{94}{7}, \frac{215}{21})$

Thus, $21 (α + β) = 497$

Q 46 :

In an equilateral triangle PQR, let the vertex P be at (3, 5) and the side QR be along the line x + y = 4. If the orthocentre of the triangle PQR is $(α, β)$ , then $9 (α + β)$ is equal to: [2026]

16
27
36
48

1

2

3

4

(4)

AP is perpendicular to QR

FIGURE

Slope of QR = –1

Equation of AP is y – 5 = 1(x – 3)

$\Rightarrow y - 5 = x - 3$

$\Rightarrow x - y = - 2$

On solving, x + y = 4 and x – y = –2, we get

x = 1 and y = 3

$∴ A \equiv (1, 3)$

Now, $G \equiv (\frac{2 + 3}{3}, \frac{6 + 5}{3}) = (5 / 3, 11 / 3) = (α, β)$

$∴ 9 (α + β) = 9 (\frac{5}{3} + \frac{11}{3}) = 9 \times \frac{16}{3} = 48$

Q 47 :

If a straight line drawn through the point of intersection of the lines 4x + 3y – 1 = 0 and 3x + 4y – 1 = 0, meets the coordinate axes at the points P and Q, then the locus of the midpoint of PQ is: [2026]

x + y – 7 = 0
x + y – 14xy = 0
2x + y + 14xy = 0
x + 2y – 14xy = 0

1

2

3

4

(2)

$L_{1} : 4 x + 3 y - 1 = 0$

$L_{2} : 3 x + 4 y - 1 = 0$

$L : L_{1} + λ L_{2} = 0$

$\Rightarrow (4 x + 3 y - 1) + λ (3 x + 4 y - 1) = 0$

$\Rightarrow (4 + 3 λ) x + (3 + 4 λ) y - (1 + λ) = 0$

Point of intersection with x-axis:

$P (\frac{λ + 1}{3 λ + 4}, 0)$

Point of intersection with y-axis:

$Q (0, \frac{λ + 1}{3 + 4 λ})$

Let mid-point of PQ be (h, k)

$h = \frac{\frac{λ + 1}{3 λ + 4} + 0}{2} = \frac{λ + 1}{6 λ + 8}$ ... (i)

$k = \frac{0 + \frac{λ + 1}{3 + 4 λ}}{2} = \frac{λ + 1}{8 λ + 6}$ ... (ii)

From (i), $λ = \frac{1 - 8 h}{6 h - 1}$

Put ' $λ$ ' in eq (ii), we get

$k = \frac{\frac{1 - 8 h}{6 h - 1} + 1}{8 (\frac{1 - 8 h}{6 h - 1}) + 6}$

$\Rightarrow h + k - 14 h k = 0$

$∴$ Locus is x + y – 14xy = 0

Q 48 :

Let A, B be points on the two half-lines $x - \sqrt{3} | y | = α$ , $α > 0$ at a distance of $α$ from their point of intersection P. The line segment AB meets the angle bisector of the given half-lines at the point Q. If $P Q = \frac{9}{2}$ and R is the radius of the circumvircle of $△$ PAB, then $\frac{α^{2}}{R}$ is equal to ________. [2026]

(9)

Two half-line are $x - \sqrt{3} | y | = α$ , $α > 0$ , meeting at point P.

Points A and B are on these half-lines and each is at distance $α$ from P.

$∴$ Lines are $x - \sqrt{3} | y | = α$ , $α > 0$ , and $x + \sqrt{3} y = α (y < 0)$

Their slopes are $\frac{1}{\sqrt{3}}, - \frac{1}{\sqrt{3}}$

So, angle between them is 60°.

Since PA = PB = $α$ ,

$∴$ $△$ PAB is isosceles triangle with vertex angle 60°.

Hence, PAB is an equilateral triangle

FIGURE

$P Q = \frac{\sqrt{3} α}{2} = \frac{9}{2}$ [ $∵$ Given]

$∴ α = \frac{9}{\sqrt{3}} \Rightarrow α = 3 \sqrt{3}$

Now, $\frac{α}{2 R} = \cos 30 ° \Rightarrow α \frac{\sqrt{3}}{2} \cdot 2 R$

$\Rightarrow α = \sqrt{3} R \Rightarrow \sqrt{3} R = 3 \sqrt{3}$ [ $∵ α = 3 \sqrt{3}$ ]

$\Rightarrow R = 3$

So, $\frac{α^{2}}{R} = \frac{{(3 \sqrt{3})}^{2}}{3} = \frac{27}{3} = 9$

Q 49 :

From the point (–1, –1), two rays are sent making angles of 45° with the line x + y = 0. These rays get reflected from the mirror x + 2y = 1. If the equations of the reflected rays are ax + by = 0 and cx + dy = 7, a, b, c, d $\in$ Z, then the value of ad + bc is ________. [2026]

(7)

We have, $\tan 45 ° = | \frac{m + 1}{1 - m} |$

$\Rightarrow \pm 1 = \frac{m + 1}{1 - m}$

$\Rightarrow$ m = 0 & one line is perpendicular to x-axis.

Rays are : $L_{1} : x = - 1, L_{2} : y = - 1$

Intersection points of $L_{1}$ and $L_{2}$ with x + 2y = 1 are (–1, 1) and (3, –1) respectively.

Now, taking mirror image of x + 1 = 0 in x + 2y = 1.

$A' = \frac{x + 1}{1} = \frac{y + 1}{2} = \frac{- 2 (1 - 2 - 1)}{1^{2} + 2^{2}} = \frac{8}{5}$

$\Rightarrow x = \frac{3}{5}, y = \frac{11}{5} and A' \equiv (\frac{3}{5}, \frac{11}{5})$

Now, taking mirror image of y + 1 = 0 in x + 2y = 1.

$A'' = \frac{x + 1}{1} = \frac{y + 1}{2} = \frac{- 2 (1 - 2 - 1)}{1^{2} + 2^{2}} = \frac{8}{5}$

$\Rightarrow x = \frac{3}{5}, y = \frac{11}{5}$

$A'' \equiv (\frac{3}{5}, \frac{11}{5})$

Equation of ray passes through (3, –1) is

$(y + 1) = \frac{(\frac{11}{5} + 1)}{(\frac{3}{5} - 3)} (x - 3)$

$\Rightarrow 4 x + 3 y = 9$

Comparing 4x + 3y = 9 with ax + by = 9, we get

a = 4, b = 3

Similarly, equation of ray passes through (–1, 1) is

–3x + 4y = 7

Comparing with cx + dy = 7, we get

c = –3, d = 4

Thus ad + bc = $4 \times 4 + 3 \times (- 3) = 16 - 9 = 7$

Topic Question Set

Q 31 :

The line passing through the point of intersection of $x + y = 2$ , $x - y = 0$ and is parallel to $x + 2 y = 5$ is $a x + b y = c$ , then value of $a + b + c$ is _________.

Q 32 :

$A_{1}, A_{2}, \dots, A_{n}$ are points on the line $y = x$ lying in the positive quadrant such that $O A_{n} = n \cdot O A_{n - 1},$ $O$ being the origin. If $O A_{1} = 1$ and the coordinates of $A_{n}$ are $(2520 \sqrt{2}, 2520 \sqrt{2})$ , then the value of $n$ is _______.

Q 33 :

The area of the rhombus $A B C D$ is 24. The equation of the diagonal BD is $4 x + 3 y + 2 = 0$ and $A = (3, 2)$ . The length of the side of the rhombus is ______.

Q 34 :

In $Δ A B C$ , the equation of altitudes AM and BN are $x + 5 y - 3 = 0$ , $x + y - k = 0$ . If the altitude CL is given by $3 x - y - 1 = 0$ , then $k =$ ______.

Q 35 :

The centroid of the triangle formed by $(a, b), (b, c), (c, a)$ is the origin and $a^{3} + b^{3} + c^{3} = k a b c,$ then $k$ is _______.

Q 36 :

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:

Q 37 :

If a straight line having negative slope passing through the point P = (8, 2) meets the x-axis at A and y-axis at B respectively. Then the minimum value of OA + OB (O being origin) is _____

Q 38 :

In a $∆ A B C, A B = A C$ where $A$ is (3, 1) and the equation of the base $B C$ is $2 x + y = 4$ . Also, $B$ lies on $x + 3 y = 7$ . Then sum of coordinates of vertex $' C'$ is ______

Q 40 :

The equations of the sides AB, BC and CA of a triangle ABC are:

$2 x + y = 0, x + p y = 21 a, (a \neq 0)$ and $x - y = 3$ respectively. Let $P (2, a)$ be the centroid of $∆ A B C$ . Then ${(B C)}^{2}$ is equal to

Q 41 :

Let the circumcenter of a triangle with vertices $A (a, 3), B (b, 5)$ and $C (a, b), a b > 0$ be P(1, 1). If the line AP intersects the line BC at the point $Q (k_{1}, k_{2})$ , then $k_{1} + k_{2}$ is equal to

Q 43 :

If A(-2, -1), B(-2, 1), C(2, 1) are the vertices of a triangle ABC, then the perpendicular distance from its circumcenter to the side BC is

Q 45 :

Let the vertex A of a triangle ABC be (1, 2) and the midpoint of the side AB be(5, –1). If the centroid of this triangle is (3, 4) and its circumcenter is $(α, β)$ , then $21 (α + β)$ is equal to: [2026]

Q 46 :

In an equilateral triangle PQR, let the vertex P be at (3, 5) and the side QR be along the line x + y = 4. If the orthocentre of the triangle PQR is $(α, β)$ , then $9 (α + β)$ is equal to: [2026]

Q 47 :

If a straight line drawn through the point of intersection of the lines 4x + 3y – 1 = 0 and 3x + 4y – 1 = 0, meets the coordinate axes at the points P and Q, then the locus of the midpoint of PQ is: [2026]

Q 49 :

From the point (–1, –1), two rays are sent making angles of 45° with the line x + y = 0. These rays get reflected from the mirror x + 2y = 1. If the equations of the reflected rays are ax + by = 0 and cx + dy = 7, a, b, c, d $\in$ Z, then the value of ad + bc is ________. [2026]

Want Us to Email you About Special Offers & Updates?

Topic Question Set

Q 31 : The line passing through the point of intersection of x+y=2, x-y=0 and is parallel to x+2y=5 is ax+by=c, then value of a+b+c is _________.

Q 32 : A1,A2,…,An are points on the line y=x lying in the positive quadrant such that OAn=n·OAn-1, O being the origin. If OA1=1 and the coordinates of An are (25202, 25202), then the value of n is _______.

Q 33 : The area of the rhombus ABCD is 24. The equation of the diagonal BD is 4x+3y+2=0 and A=(3, 2). The length of the side of the rhombus is ______.

Q 34 : In ΔABC, the equation of altitudes AM and BN are x+5y-3=0, x+y-k=0. If the altitude CL is given by 3x-y-1=0, then k= ______.

Q 35 : The centroid of the triangle formed by (a,b), (b,c), (c,a) is the origin and a3+b3+c3=kabc, then k is _______.

Q 36 : 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:

Q 37 : If a straight line having negative slope passing through the point P = (8, 2) meets the x-axis at A and y-axis at B respectively. Then the minimum value of OA + OB (O being origin) is _____

Q 38 : In a ∆ABC, AB=AC where A is (3, 1) and the equation of the base BC is 2x+y=4. Also, B lies on x+3y=7. Then sum of coordinates of vertex 'C' is ______

Q 40 : The equations of the sides AB, BC and CA of a triangle ABC are: 2x+y=0, x+py=21a, (a≠0) and x-y=3 respectively. Let P(2,a) be the centroid of ∆ABC. Then (BC)2 is equal to

Q 41 : Let the circumcenter 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 Q(k1,k2), then k1+k2 is equal to

Q 42 : Consider the following statements: S1: For the line 2x+3y-7=0, the points origin and (1, 1) are on the same side. S2: Two points (x1,y1) and (x2,y2) lie on the same side of the line ax+by+c=0 (a,b,c∈R-{0}) if (ax1+by1+c)(ax2+by2+c)>0

Q 43 : If A(-2, -1), B(-2, 1), C(2, 1) are the vertices of a triangle ABC, then the perpendicular distance from its circumcenter to the side BC is

Q 44 : Let the line L1:x+3=0 intersect the lines L2:x–y=0 and L3:3x+y=0 at the points A and B, respectively. Let the bisector of the obtuse angle between the lines L2 and L3 intersect the line L1 at the point C. Then BC2:AC2 is equal to: [2026]

Q 45 : Let the vertex A of a triangle ABC be (1, 2) and the midpoint of the side AB be(5, –1). If the centroid of this triangle is (3, 4) and its circumcenter is (α,β), then 21(α+β) is equal to: [2026]

Q 46 : In an equilateral triangle PQR, let the vertex P be at (3, 5) and the side QR be along the line x + y = 4. If the orthocentre of the triangle PQR is (α,β), then 9(α+β) is equal to: [2026]

Q 47 : If a straight line drawn through the point of intersection of the lines 4x + 3y – 1 = 0 and 3x + 4y – 1 = 0, meets the coordinate axes at the points P and Q, then the locus of the midpoint of PQ is: [2026]

Q 49 : From the point (–1, –1), two rays are sent making angles of 45° with the line x + y = 0. These rays get reflected from the mirror x + 2y = 1. If the equations of the reflected rays are ax + by = 0 and cx + dy = 7, a, b, c, d∈Z, then the value of ad + bc is ________. [2026]

Want Us to Email you About Special Offers & Updates?

Q 31 :

The line passing through the point of intersection of $x + y = 2$ , $x - y = 0$ and is parallel to $x + 2 y = 5$ is $a x + b y = c$ , then value of $a + b + c$ is _________.

Q 32 :

$A_{1}, A_{2}, \dots, A_{n}$ are points on the line $y = x$ lying in the positive quadrant such that $O A_{n} = n \cdot O A_{n - 1},$ $O$ being the origin. If $O A_{1} = 1$ and the coordinates of $A_{n}$ are $(2520 \sqrt{2}, 2520 \sqrt{2})$ , then the value of $n$ is _______.

Q 33 :

The area of the rhombus $A B C D$ is 24. The equation of the diagonal BD is $4 x + 3 y + 2 = 0$ and $A = (3, 2)$ . The length of the side of the rhombus is ______.

Q 34 :

In $Δ A B C$ , the equation of altitudes AM and BN are $x + 5 y - 3 = 0$ , $x + y - k = 0$ . If the altitude CL is given by $3 x - y - 1 = 0$ , then $k =$ ______.

Q 35 :

The centroid of the triangle formed by $(a, b), (b, c), (c, a)$ is the origin and $a^{3} + b^{3} + c^{3} = k a b c,$ then $k$ is _______.

Q 36 :

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:

Q 37 :

If a straight line having negative slope passing through the point P = (8, 2) meets the x-axis at A and y-axis at B respectively. Then the minimum value of OA + OB (O being origin) is _____

Q 38 :

In a $∆ A B C, A B = A C$ where $A$ is (3, 1) and the equation of the base $B C$ is $2 x + y = 4$ . Also, $B$ lies on $x + 3 y = 7$ . Then sum of coordinates of vertex $' C'$ is ______

Q 40 :

The equations of the sides AB, BC and CA of a triangle ABC are:

$2 x + y = 0, x + p y = 21 a, (a \neq 0)$ and $x - y = 3$ respectively. Let $P (2, a)$ be the centroid of $∆ A B C$ . Then ${(B C)}^{2}$ is equal to

Q 41 :

Let the circumcenter of a triangle with vertices $A (a, 3), B (b, 5)$ and $C (a, b), a b > 0$ be P(1, 1). If the line AP intersects the line BC at the point $Q (k_{1}, k_{2})$ , then $k_{1} + k_{2}$ is equal to

Q 43 :

If A(-2, -1), B(-2, 1), C(2, 1) are the vertices of a triangle ABC, then the perpendicular distance from its circumcenter to the side BC is

Q 45 :

Let the vertex A of a triangle ABC be (1, 2) and the midpoint of the side AB be(5, –1). If the centroid of this triangle is (3, 4) and its circumcenter is $(α, β)$ , then $21 (α + β)$ is equal to: [2026]

Q 46 :

In an equilateral triangle PQR, let the vertex P be at (3, 5) and the side QR be along the line x + y = 4. If the orthocentre of the triangle PQR is $(α, β)$ , then $9 (α + β)$ is equal to: [2026]

Q 47 :

If a straight line drawn through the point of intersection of the lines 4x + 3y – 1 = 0 and 3x + 4y – 1 = 0, meets the coordinate axes at the points P and Q, then the locus of the midpoint of PQ is: [2026]

Q 49 :

From the point (–1, –1), two rays are sent making angles of 45° with the line x + y = 0. These rays get reflected from the mirror x + 2y = 1. If the equations of the reflected rays are ax + by = 0 and cx + dy = 7, a, b, c, d $\in$ Z, then the value of ad + bc is ________. [2026]