Properties of Triangle

In a $\Delta PQR,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu}$ If $3{\mkern 1mu} \sin {\mkern 1mu} P + 4{\mkern 1mu} \cos {\mkern 1mu} Q = 6$ and $4\sin Q + 3\cos P = 1,$ then the angle R is equal to :

In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y. If x² – c² = y, where c is the length of the third side of the triangle, then the circumradius of the triangle is :

A triangle ABC lying in the first quadrant has two vertices as A(1, 2) and B(3, 1). If $\angle BAC = {90^o}$ and area$\left( {\Delta ABC} \right) = 5\sqrt 5$ s units, then the abscissa of the vertex C is :

The triangle of maximum area that can be inscribed in a given circle of radius 'r' is :

The lengths of the sides of a triangle are 10 + x², 10 + x² and 20 $-$ 2x². If for x = k, the area of the triangle is maximum, then 3k² is equal to :

A straight line cuts off the intercepts $\mathrm{OA}=\mathrm{a}$ and $\mathrm{OB}=\mathrm{b}$ on the positive directions of $x$-axis and $y$ axis respectively. If the perpendicular from origin $O$ to this line makes an angle of $\frac{\pi}{6}$ with positive direction of $y$-axis and the area of $\triangle \mathrm{OAB}$ is $\frac{98}{3} \sqrt{3}$, then $\mathrm{a}^{2}-\mathrm{b}^{2}$ is equal to :

Let $\left(5, \frac{a}{4}\right)$ be the circumcenter of a triangle with vertices $\mathrm{A}(a,-2), \mathrm{B}(a, 6)$ and $C\left(\frac{a}{4},-2\right)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha+\beta+\gamma$ is

Two vertices of a triangle $\mathrm{ABC}$ are $\mathrm{A}(3,-1)$ and $\mathrm{B}(-2,3)$, and its orthocentre is $\mathrm{P}(1,1)$. If the coordinates of the point $\mathrm{C}$ are $(\alpha, \beta)$ and the centre of the of the circle circumscribing the triangle $\mathrm{PAB}$ is $(\mathrm{h}, \mathrm{k})$, then the value of $(\alpha+\beta)+2(\mathrm{~h}+\mathrm{k})$ equals

In a triangle with sides $a, b, c,$ ${r_1} > {r_2} > {r_3}$ (which are the ex-radii) then :