Consider the set of all triangles OPQ where O | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The points $P$ and $Q$ lie on the line $5x + y = 99$. Since $x, y \ge 0$ and are integers,$x$ can take values from $0$ to $19$ (as $5(19) + y = 99

Vedclass · Accepted Answer

$90$

Vedclass · Answer

(C) The points $P$ and $Q$ lie on the line $5x + y = 99$. Since $x, y \ge 0$ and are integers,$x$ can take values from $0$ to $19$ (as $5(19) + y = 99 \implies y = 4$). There are $20$ such points.Let $P = (x_1, 99 - 5x_1)$ and $Q = (x_2, 99 - 5x_2)$.The area of $\Delta OPQ$ is given by $\Delta = \frac{1}{2} |x_1(99 - 5x_2) - x_2(99 - 5x_1)|$.$\Delta = \frac{1}{2} |99x_1 - 5x_1x_2 - 99x_2 + 5x_1x_2| = \frac{99}{2} |x_1 - x_2|$.For the area to be an integer,$|x_1 - x_2|$ must be even.This means $x_1$ and $x_2$ must have the same parity (both even or both odd).There are $10$ even values $(0, 2, \dots, 18)$ and $10$ odd values $(1, 3, \dots, 19)$.The number of ways to choose two distinct points is $^{10}C_2 + ^{10}C_2 = 45 + 45 = 90$.

Consider the set of all triangles $OPQ$ where $O$ is the origin and $P, Q$ are distinct points in the plane with non-negative integral coordinates $(x, y)$ such that $5x + y = 99$. The number of such distinct triangles whose area is a positive integer is:

Similar Questions

If the sides $AB, BC, CD$ and $DA$ of a quadrilateral are given by the equations $x + 2y = 3, x = 1, x - 3y = 4$ and $5x + y + 12 = 0$ respectively,then find the angle between the diagonals $AC$ and $BD$ in degrees.

The area (in sq units) of the quadrilateral formed by the lines $2x + 3y + 6 = 0$,$2x - 3y + 6 = 0$,$2x + 3y - 6 = 0$,and $2x - 3y - 6 = 0$ is

If one of the diagonals of a square is along the line $x = 2y$ and one of its vertices is $(3, 0)$,then the equations of the sides passing through this vertex are:

The area of the triangle formed by the lines $y = m_1x + c_1$,$y = m_2x + c_2$ and $x = 0$ is:

The circumcentre of the triangle formed by the lines $xy+2x+2y+4=0$ and $x+y+2=0$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module