A line meets the coordinate axes at A and B. | vedclass

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

Question

(C) Let the line $\frac{x}{a} + \frac{y}{b} = 1$ meet the coordinate axes at $A(a, 0)$ and $B(0, b)$.Since $	riangle OAB$ is a right-angled triangle,

Vedclass · Accepted Answer

$d_1 + d_2$

Vedclass · Answer

(C) Let the line $\frac{x}{a} + \frac{y}{b} = 1$ meet the coordinate axes at $A(a, 0)$ and $B(0, b)$.Since $	riangle OAB$ is a right-angled triangle,the circumcircle has the hypotenuse $AB$ as its diameter.The center of the circle is $C = (\frac{a}{2}, \frac{b}{2})$ and the radius is $r = \frac{\sqrt{a^2 + b^2}}{2}$.The slope of $OC$ is $\frac{b}{a}$,so the slope of the tangent at the origin $O$ is $-\frac{a}{b}$.The equation of the tangent at $O$ is $ax + by = 0$.The distance $d_1$ from $A(a, 0)$ to the tangent $ax + by = 0$ is $d_1 = \frac{|a(a) + b(0)|}{\sqrt{a^2 + b^2}} = \frac{a^2}{\sqrt{a^2 + b^2}}$.The distance $d_2$ from $B(0, b)$ to the tangent $ax + by = 0$ is $d_2 = \frac{|a(0) + b(b)|}{\sqrt{a^2 + b^2}} = \frac{b^2}{\sqrt{a^2 + b^2}}$.Adding these distances,we get $d_1 + d_2 = \frac{a^2 + b^2}{\sqrt{a^2 + b^2}} = \sqrt{a^2 + b^2}$.Since the diameter is $2r = \sqrt{a^2 + b^2}$,the diameter is $d_1 + d_2$.

$A$ line meets the coordinate axes at $A$ and $B$. $A$ circle is circumscribed about the triangle $OAB$. If $d_1$ and $d_2$ are the distances of the tangent to the circle at the origin $O$ from the points $A$ and $B$ respectively,then the diameter of the circle is:

Similar Questions

From a point $A(0,3)$ on the circle $(x+2)^2+(y-3)^2=4$,a chord $AB$ is drawn and it is extended to a point $Q$ such that $AQ=2AB$. Then the locus of $Q$ is

If a tangent to the circle $x^2 + y^2 = 1$ intersects the coordinate axes at distinct points $P$ and $Q,$ then the locus of the mid-point of $PQ$ is

If the tangents to the circle $x^2 + y^2 = a^2$ with slopes $\alpha$ and $\beta$ intersect at point $P$,and $\cot \alpha + \cot \beta = 0$,then the locus of $P$ is:

The equation of the image of the circle $x^2 + y^2 + 16x - 24y + 183 = 0$ by the line mirror $4x + 7y + 13 = 0$ is:

If the distance of a variable point $P(x, y)$ from a point $A(2, -2)$ is twice the distance of $P$ from the $Y$-axis,then the equation of the locus of $P$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module