If the tangent to the circle x^2 + y^2 = r^2 | vedclass

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

Question

(A) The equation of the tangent to the circle $x^2 + y^2 = r^2$ at the point $(a, b)$ is given by $ax + by = r^2$.To find the points where the tangent

Vedclass · Accepted Answer

$\frac{r^4}{2ab}$

Vedclass · Answer

(A) The equation of the tangent to the circle $x^2 + y^2 = r^2$ at the point $(a, b)$ is given by $ax + by = r^2$.To find the points where the tangent meets the coordinate axes,we rewrite the equation in intercept form:$\frac{ax}{r^2} + \frac{by}{r^2} = 1 \Rightarrow \frac{x}{r^2/a} + \frac{y}{r^2/b} = 1$.Thus,the coordinates of $A$ (on the $y$-axis) and $B$ (on the $x$-axis) are $(0, r^2/b)$ and $(r^2/a, 0)$ respectively.The lengths of the intercepts are $OA = |r^2/b|$ and $OB = |r^2/a|$.The area of the right-angled triangle $OAB$ is given by $\frac{1}{2} 	imes OA 	imes OB = \frac{1}{2} 	imes \left|\frac{r^2}{a}ight| 	imes \left|\frac{r^2}{b}ight| = \frac{r^4}{2|ab|}$.Assuming $a$ and $b$ are positive as per the diagram,the area is $\frac{r^4}{2ab}$.

If the tangent to the circle $x^2 + y^2 = r^2$ at the point $(a, b)$ meets the coordinate axes at the points $A$ and $B$,and $O$ is the origin,then the area of the triangle $OAB$ is

Similar Questions

Consider the two curves $C_1: y^2=4x$ and $C_2: x^2+y^2-6x+1=0$. Then,

The number of real circles cutting orthogonally the circle $x^{2}+y^{2}+2x-2y+7=0$ is

If the line $ax + by = 0$ touches the circle ${x^2} + {y^2} + 2x + 4y = 0$ and is a normal to the circle ${x^2} + {y^2} - 4x + 2y - 3 = 0$,then the value of $(a, b)$ will be

The centre$(s)$ of the circle$(s)$ passing through the points $(0, 0)$ and $(1, 0)$ and touching the circle $x^2 + y^2 = 9$ is/are:

The sum of diameters of the circles that touch $(i)$ the parabola $75x^2 = 64(5y - 3)$ at the point $\left(\frac{8}{5}, \frac{6}{5}\right)$ and $(ii)$ the $y$-axis,is equal to $......$

Vedclass Test Series

Exam Paper Generator

Online Exam Module