In radius of a circle which is inscribed in a isosceles
triangle one of whose angle is $ 2 \pi / 3, \, is \, \sqrt 3 $, then area
of triangle (in sq units) is
- 4 $ \sqrt 3 $
- 12 - 7 $ \sqrt 3 $
- 12 + 7 $ \sqrt 3 $
- None of these
The Correct Option is C
Solution and Explanation
Top Questions on Circle
- If $ r_1 $ and $ r_2 $ are radii of two circles touching all the four circles $$ (x \pm r)^2 + (y \pm r)^2 = r^2, $$ then find the value of $$ \frac{r_1 + r_2}{r}. $$
- If the equation of the circle having the common chord to the circles $$ x^2 + y^2 + x - 3y - 10 = 0 $$ and $$ x^2 + y^2 + 2x - y - 20 = 0 $$ as its diameter is $$ x^2 + y^2 + \alpha x + \beta y + \gamma = 0, $$ then find $ \alpha + 2\beta + \gamma $.
- The circles $$ x^2 + y^2 - 2x - 4y - 4 = 0 $$ and $$ x^2 + y^2 + 2x + 4y - 11 = 0 $$ ...
- The slope of the common tangent drawn to the circles $$ x^2 + y^2 - 4x + 12y - 216 = 0 $$ and $$ x^2 + y^2 + 6x - 12y + 36 = 0 $$ is:
- If the line $$ 4x - 3y + 7 = 0 $$ touches the circle $$ x^2 + y^2 - 6x + 4y - 12 = 0 $$ at $ (\alpha, \beta) $, then find $ \alpha + 2\beta $.
Questions Asked in JEE Advanced exam
- Let $ x_0 $ be the real number such that $ e^{x_0} + x_0 = 0 $. For a given real number $ \alpha $, define $$ g(x) = \frac{3xe^x + 3x - \alpha e^x - \alpha x}{3(e^x + 1)} $$ for all real numbers $ x $. Then which one of the following statements is TRUE?
- JEE Advanced - 2025
- Fundamental Theorem of Calculus
- At 25$^\circ$C, the concentration of H$^+$ ions in $ 1.00 \times 10^{-3} \, \text{M} $ aqueous solution of a weak monobasic acid having acid dissociation constant $ K_a = 4.00 \times 10^{-11} $ is $ X \times 10^{-7} \, \text{M} $. The value of $ X $ is ____. {Use: Ionic product of water $ K_w = 1.00 \times 10^{-14} $ at 25$^\circ$C
- JEE Advanced - 2025
- Ionic Equilibrium
- Consider the vectors $$ \vec{x} = \hat{i} + 2\hat{j} + 3\hat{k},\quad \vec{y} = 2\hat{i} + 3\hat{j} + \hat{k},\quad \vec{z} = 3\hat{i} + \hat{j} + 2\hat{k}. $$ For two distinct positive real numbers $ \alpha $ and $ \beta $, define $$ \vec{X} = \alpha \vec{x} + \beta \vec{y} - \vec{z},\quad \vec{Y} = \alpha \vec{y} + \beta \vec{z} - \vec{x},\quad \vec{Z} = \alpha \vec{z} + \beta \vec{x} - \vec{y}. $$ If the vectors $ \vec{X}, \vec{Y}, \vec{Z} $ lie in a plane, then the value of $ \alpha + \beta - 3 $ is ________.
Two identical concave mirrors each of focal length $ f $ are facing each other as shown. A glass slab of thickness $ t $ and refractive index $ n_0 $ is placed equidistant from both mirrors on the principal axis. A monochromatic point source $ S $ is placed at the center of the slab. For the image to be formed on $ S $ itself, which of the following distances between the two mirrors is/are correct:
- JEE Advanced - 2025
- Ray optics and optical instruments
- Let $$ \alpha = \frac{1}{\sin 60^\circ \sin 61^\circ} + \frac{1}{\sin 62^\circ \sin 63^\circ} + \cdots + \frac{1}{\sin 118^\circ \sin 119^\circ}. $$ Then the value of $$ \left( \frac{\csc 1^\circ}{\alpha} \right)^2 $$ is \rule{1cm}{0.15mm}.
- JEE Advanced - 2025
- Some Applications of Trigonometry
Concepts Used:
Circle
A circle can be geometrically defined as a combination of all the points which lie at an equal distance from a fixed point called the centre. The concepts of the circle are very important in building a strong foundation in units likes mensuration and coordinate geometry. We use circle formulas in order to calculate the area, diameter, and circumference of a circle. The length between any point on the circle and its centre is its radius.
Any line that passes through the centre of the circle and connects two points of the circle is the diameter of the circle. The radius is half the length of the diameter of the circle. The area of the circle describes the amount of space that is covered by the circle and the circumference is the length of the boundary of the circle.
Also Check: