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)

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

$\Rightarrow a + b + c = 0$

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

$\Rightarrow k = 3$

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 _______.

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 _______.

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 _______.