A circle touching the x- axis at (3, 0) and m | vedclass

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

Question

(C) Let the center of the circle be $(3, r)$ or $(3, -r)$ and the radius be $r$.Since the circle touches the $x-$ axis at $(3, 0)$,its equation is $(x

Vedclass · Accepted Answer

$(3, 10)$

Vedclass · Answer

(C) Let the center of the circle be $(3, r)$ or $(3, -r)$ and the radius be $r$.Since the circle touches the $x-$ axis at $(3, 0)$,its equation is $(x - 3)^2 + (y - r)^2 = r^2$ or $(x - 3)^2 + (y + r)^2 = r^2$.Expanding this,we get $x^2 - 6x + 9 + y^2 \mp 2ry + r^2 = r^2$,which simplifies to $x^2 + y^2 - 6x \mp 2ry + 9 = 0$.The $y-$ intercept is given by $2\sqrt{f^2 - c}$,where $f = \mp r$ and $c = 9$.Given the intercept length is $8$,we have $2\sqrt{r^2 - 9} = 8$.$\sqrt{r^2 - 9} = 4 \implies r^2 - 9 = 16 \implies r^2 = 25 \implies r = 5$.The equations of the circles are $x^2 + y^2 - 6x + 10y + 9 = 0$ and $x^2 + y^2 - 6x - 10y + 9 = 0$.Checking the point $(3, 10)$ in the second equation: $(3)^2 + (10)^2 - 6(3) - 10(10) + 9 = 9 + 100 - 18 - 100 + 9 = 0$.Thus,the circle passes through $(3, 10)$.

$A$ circle touching the $x-$ axis at $(3, 0)$ and making an intercept of length $8$ on the $y-$ axis passes through the point:

Similar Questions

Let the mirror image of a circle $c_{1}: x^{2}+y^{2}-2x-6y+\alpha=0$ in the line $y=x+1$ be $c_{2}: 5x^{2}+5y^{2}+10gx+10fy+38=0$. If $r$ is the radius of circle $c_{2}$,then $\alpha+6r^{2}$ is equal to:

For what values of $m$ does the circle $x^2 + y^2 = 4x + 8y + 5$ intersect the line $3x - 4y = m$ at two distinct points?

Let $ABCD$ be a quadrilateral with area $18$,with side $AB$ parallel to the side $CD$ and $AB = 2CD$. Let $AD$ be perpendicular to $AB$ and $CD$. If a circle is drawn inside the quadrilateral $ABCD$ touching all the sides,then its radius is

The equation of the pair of straight lines parallel to the $x$-axis and touching the circle $x^2+y^2-6x-4y-12=0$ is

The length of the chord intercepted by the circle $x^2+y^2-4x+4y+3=0$ on the line $x=3y+13$ is units.

Vedclass Test Series

Exam Paper Generator

Online Exam Module