Home
Class 12
MATHS
Prove that the line y=m(x-1)+3sqrt(1+m^(...

Prove that the line `y=m(x-1)+3sqrt(1+m^(2))-2` touches the circle `x^(2)+y^(2)-2x+4y-4=0` for all reacl values of m.

Text Solution

Verified by Experts

The equation of given circle is `(x-1)^(2)+(y+2)^(2)=3^(2)` .
Therefore, equations of tangents to the circle having slope m are
`y+2=m(x-1)+-3 sqrt(1+m^(2))`
Alternatively,
As any tangent to `x^(2)+y^(2)=3^(2)` is given by `y=mx+3sqrt(1+m^(2))` , any tangent to the given circle will be
`y+2=m(x-1)+3 sqrt(1+m^(2))`
Promotional Banner

Topper's Solved these Questions

Similar Questions

Explore conceptually related problems

Determine whether x+y-1=0 is the equation of a diameter of the circle x^(2)+y^(2)-6x+4y+c=0 for all possible values of c.

The line y=m x-((a^2-b^2)m)/(sqrt(a^2+b^2m^2)) is normal to the ellise (x^2)/(a^2)+(y^2)/(b^2)=1 for all values of m belonging to (a) (0,1) (b) (0,oo) (c) R (d) none of these

Find the image of the circle x^2+y^2-2x+4y-4=0 in the line 2x-3y+5=0

If the line k^(2)(x-1)+k(y-2)+1=0 touches the parabola y^(2)-4x-4y+8=0 , then k can be

The values of m for which the lines y = mx + 2 sqrt5 touches the hyperbola 16 x^(2) - 9y^(2) = 144 are the roots of x^(2) - (a+b) x-4 = 0 then the value of (a+b) is

A circle touching the line x +y - 2 = 0 at (1,1) and cuts the circle x^(2) +y^(2) +4x +5y - 6 = 0 at P and Q. Then

A variable circle which always touches the line x+y-2=0 at (1, 1) cuts the circle x^2+y^2+4x+5y-6=0 . Prove that all the common chords of intersection pass through a fixed point. Find that points.

The circle x^2 + y^2 - 4x + 6y + c = 0 touches x axis if

Prove that the equation of any tangent to the circle x^2+y^2-2x+4y-4=0 is of the form y=m(x-1)+3sqrt(1+m^2)-2.