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

$7x+y-24=0$ and $x+7y-24=0$ represent the equal sides of an isosceles triangle. If the third side passes through $(-1, 1)$,then a possible equation for the third side is

The triangle formed by the lines $x + y - 4 = 0,$ $3x + y = 4,$ and $x + 3y = 4$ is

For what value of $k$ are the points $(k, 2 - 2k)$,$(1 - k, 2k)$,and $(-4 - k, 6 - 2k)$ collinear?

If $A(1,3)$ and $C(7,5)$ are two opposite vertices of a square,then find the equation of a side passing through $A$.

$A$ straight line $4x+y-1=0$ passing through the point $A(2,-7)$ meets the line $BC$ whose equation is $3x-4y+1=0$ at the point $B$. Then the equation of the line $AC$ such that $AB=AC$,is

Vedclass Test Series

Exam Paper Generator

Online Exam Module