If x^2 + y^2 + 2gx + 2fy + c = 0 is the equat | vedclass

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

Question

(A) For the smallest circle,the line segment connecting the point $(1, 2)$ and its projection on the line $x + y - 7 = 0$ acts as the diameter.Let the

Vedclass · Accepted Answer

$25$

Vedclass · Answer

(A) For the smallest circle,the line segment connecting the point $(1, 2)$ and its projection on the line $x + y - 7 = 0$ acts as the diameter.Let the point on the line be $B(h, k)$. The line segment from $(1, 2)$ to $B$ is perpendicular to $x + y - 7 = 0$ (slope $-1$).Thus,the slope of the line segment is $1$. So,$\frac{k - 2}{h - 1} = 1 \implies k - 2 = h - 1 \implies k = h + 1$.Since $B$ lies on $x + y - 7 = 0$,we have $h + (h + 1) - 7 = 0 \implies 2h = 6 \implies h = 3$. Then $k = 4$.So,the diameter endpoints are $(1, 2)$ and $(3, 4)$.The equation of the circle is $(x - 1)(x - 3) + (y - 2)(y - 4) = 0$.$x^2 - 4x + 3 + y^2 - 6y + 8 = 0 \implies x^2 + y^2 - 4x - 6y + 11 = 0$.Comparing with $x^2 + y^2 + 2gx + 2fy + c = 0$,we get $2g = -4 \implies g = -2$,$2f = -6 \implies f = -3$,and $c = 11$.The value of $(g + 2f + 3c) = -2 + 2(-3) + 3(11) = -2 - 6 + 33 = 25$.

If $x^2 + y^2 + 2gx + 2fy + c = 0$ is the equation of the smallest circle passing through $(1, 2)$ and touching the line $x + y - 7 = 0$,then the value of $(g + 2f + 3c)$ is -

Similar Questions

The image of the point $(3, 4)$ with respect to the radical axis of the circles $x^2 + y^2 + 8x + 2y + 10 = 0$ and $x^2 + y^2 + 7x + 3y + 10 = 0$ is

The equations of the circles, which touch both the axes and the line $4x+3y=12$ and have centers in the first quadrant, are

If a circle with radius $2.5$ units passes through the points $(2, 3)$ and $(5, 7)$,then its centre is

If the length of the chord $2x+3y+k=0$ of the circle $x^2+y^2-6x-8y+9=0$ is $2\sqrt{3}$,then one of the values of $k$ is

Which of the following statements are true and which are false? In each case,give a valid reason for your answer.
$q$: The centre of a circle bisects each chord of the circle.

Vedclass Test Series

Exam Paper Generator

Online Exam Module