Quiz on circle

Conditions of the tangency of the line  $y=mx+c$  to the circle  $x^2+y^2=a^2.$  

• length of the perpendicular from the centre to the line equal to the radius of the circle.

• length of the perpendicular from the center to the line is not equal to the radius of the circle.

• length of the perpendicular from the point of contact to the line equal to the radius of the circle.

• length of the perpendicular to the line equal to the radius of the circle.

Equation of the tangent at  $\left(x_1,y_1\right)$  to the circle  $x^2+y^2=a^2$  is

•  $xx_1-yy_1=a^2$  

•  $xx_1+yy_1=a^2$  

•  $xx_1+yy_1=r^2$  

•  $xx_1+yy_1+a^2=0$  

Equation of the tangent at  $\left(x_1,\ y_1\right)$  to the circle  $x^2+y^2+2gx+2fy+c=0$  

•  $xx_1+yy_1=a^2$  

•  $xx_1-yy_1+2gx+2fy=o$  

•  $xx_1-yy_1+f\left(x+x_1\right)-g\left(y+y_1\right)-c=0$  

•  $xx_1+yy_1+g\left(x+x_1\right)+f\left(y+y_1\right)+c=0$  

Length of the tangent at  $\left(x_1,\ y_1\right)$  to the circle  $x^2+y^2+2gx+2fy+c=0$  

•  $xx_1+yy_1=a^2$  

•  $\sqrt{x_1^2+y_1^2+2gx_1+2fy_1-c}$  

•  $x_1^2+y_1^2+2gx_1-2fy_1+c=0$  

•  $xx_1-yy_1+2gx_1+2fy_1+c=0$  

Find the value of k so that the length of the tangent from (5,4) to the circle  $x^2+y^2+2ky=0$  is 1.

• 0

• -5

• 2

• 1

Find the equation of the circle with centre at (4,5) and  $3x-4y+5=0$  as the line tangent to the circle.

•  $x^2+y^2-3x=0$  

•  $x^2+y^2-3x+4y=0$  

•  $x^2+y^2-5x+7y-7=0$  

•  $25x^2+25y^2+-200x-250y+1016=0$  

Find the value of k so that the straight line  $4x+3y+k=0$  may touch the circle  $x^2+y^2-4x+10y+4=0$ . Hints: Please use the concept that the Length of the perpendicular from the centre of the circle to the line equal to the radius of the circle.

•  $38,\ -18$  

•  $1,\ 3$  

•  $2,0$  

•  $4,5$  

What is the condition that  $lx+my+n=0$  may be a tangent to the circle  $x^2+y^2+2gx+2fy+c=0$ .

•  $\left(n-\lg-mf\right)^2=\left(l^2+m^2\right)\left(g^2+f^2-c\right)$  

•  $\lg-mf-n=o$  

•  $\ln-mf-g=\left(g^2+f^2-c\right)$  

•  $lm-mn-p=g^2+f^2-c^2$  

Find the equation of the tangent to the circle  $x^2+y^2-2x-4y-4=0$  which are perpendicular to  $3x-4y=14.$  

•  $4x+3y+5=0,\ 4x+3y-25=0$  

•  $4x-5y+2=0$  

•  $3x-4y+2=0$  

•  $3x-4y+2=0,\ 5x-2y+6=0$  

Find the equation of the circle through the points (1,2), (3,1) and (-3, -1).

• $x^2+y^2-2x+3y-5=0$

• $x^2+y^2-x+3y-10=0$

• $x^2-y^2+4x=5y+8=0$

• $x^2-y^2+5x-27y+5=0$