Question

Equation of a line is $y=2x.$
a) A is a point on the line. If the x coordinate of A is $-2$, find its y coordinate.
b) Verify whether a circle of radius 5 centred at A passes through the point $B(5,5).$
c) Radius of a circle passing through B is 5 and its centre is on the above mentioned line. Find the coordinates of its centre.

Answer 1

Equation of the line is y = 2x
(a) Point A lies on the line y = 2x so it satisfies the equation of the line.
$y=2(-2)=-4$
Hence y coordinate of point A is -4.
(b) To verify whether a circle of radius 5 centred at point A(-2, -4) passes through the point B(5, 5) it is enough to show that the distance between point A and B is 5.
$AB=\sqrt{(5+2{)}^2+(5+4{)}^2}$
$=\sqrt{49+81}$
$=\sqrt{130}$
$\ne 5$
Hence a circle of radius 5 centred at point A(-2, -4) does not pass through the point B(5, 5).
(c) Let $(x_1,y_1)$ be the coordinates of centre C on line.
Therefore it satisfies the equation of line a y = 2x
$\therefore y_1=2x_1$
Also the distane between point B and C is radius that is 5.
So $B{C}^2=25$
$(x_1-5{)}^2+(y_1-5{)}^2=25$
$(x_1-5{)}^2+(2x_1-5{)}^2=25$ $[\because y_1=2x_1]$
$5x_1^2-30x_1+25=0$
$x_1^2-6x_1+5=0$
$(x_1-5)(x_1-1)=0$
$x_1=5$ or $1$
$y_1=2x_1=2\left(5\right)=10$
$y_1=2x_1=2\left(1\right)=2$
The coordinates of point of C are (5, 10) or (1, 2).