Area of Triangle in Coordinate Geometry - Practice Questions & MCQ

Let a line $y=m x(m>0)$ intersect the parabola, $y^2=x_{\text {at a point } \mathrm{P} \text {, other than the }}$ origin. Let the tangent to it at P meet the x-axis at the point Q . If area $(\triangle O P Q)=4$ sq.units, then m is equal to $\qquad$

Drawn from the origin are two mutually perpendicular lines forming an isosceles triangle together with a straight line , then the area of this triangle is:

Let three vectors $\overrightarrow{\mathrm{a}}=\alpha \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$, $\vec{b}=5 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{c}=x \hat{i}+y \hat{j}+z \hat{k}$ from a triangle such that $\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}$ and the area of the triangle is $5 \sqrt{6}$. if $\alpha$ is a positive real number, then $|\overrightarrow{\mathrm{c}}|^2$ is :

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $(0, 0)$, $(0, 41)$ and $(41, 0)$ is :