Home
Class 12
MATHS
The straight line x+2y=1 meets the coord...

The straight line x+2y=1 meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then the sum of perpendicular distances from A and B on the tangent to the circle at the origin is

A

`2sqrt5`

B

`(sqrt5)/(4)`

C

`4sqrt5`

D

`(sqrt5)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
D
According to given information, we have the following figure.

From figure, equation of circle (diameter form) is
`(x-1)(x-0)+(y-0)((y-(1)/(2))=0`
`rArr x^(2) + y^(2) -x -(y)/(2)=0`
Equation of tangent at (0, 0) is `x+(y)+(2) = 0`
`[therefore" equation of tangent at "(x_(1),y_(1)) " is given by " T=0 " Here ", T = 0`
`rArr "xx"_(1)+yy_(1)-(1)/(2)(x+x_(1))-(1)/(4)(y+y_(1))=0]`
`rArr 2x + y =0 `
Now, `AM=(|2.1+1.0|)/(sqrt5)=(2)/(sqrt5)`
`[therefore " distance of a point " P(x_(1), y_(1))` from a line
`ax+by + c=0 " is " (|ax_(1)+by_(1)+c|)/(sqrt(a^(2)+b^(2)))]`
and `BN = (|2.0+1((1)/(2))|)/(sqrt5)=(1)/(2sqrt5)`
`therefore AM+BN=(2)/(sqrt5)+(1)/(2sqrt5)=(4+1)/(2sqrt5)=(sqrt5)/(2)`
Promotional Banner

Similar Questions

Explore conceptually related problems

A line 2x + y = 1 intersect co-ordinate axis at points A and B . A circle is drawn passing through origin and point A & B . If perpendicular from point A and B are drawn on tangent to the circle at origin then sum of perpendicular distance is (A) 5/sqrt2 (B) sqrt5/2 (C) sqrt5/4 (D) 5/2

The line 3x+4y-12=0 meets the coordinate axes at A and B. find the equation of the circle drawn on AB as diameter.

The line 5x+3y-15=0 meets the coordinate axes at A and B. Find the equation of the circle drawn on AB as diameter.

A line meets the coordinate axes at A and B . A circle is circumscribed about the triangle O A Bdot If d_1a n dd_2 are distances of the tangents to the circle at the origin O from the points Aa n dB , respectively, then the diameter of the circle is (2d_1+d_2)/2 (b) (d_1+2d_2)/2 d_1+d_2 (d) (d_1d_2)/(d_1+d_2)

The tangent at any point P on the circle x^2+y^2=4 meets the coordinate axes at Aa n dB . Then find the locus of the midpoint of A Bdot

A variable chord is drawn through the origin to the circle x^2+y^2-2a x=0 . Find the locus of the center of the circle drawn on this chord as diameter.

A line is drawn through the point (1, 2) to meet the coordinate axes at P and Q such that it forms a triangle OPQ, where O is the origin. If the area of the triangle OPQ is least, then the slope of the line PQ is

The distance between the origin and the tangent to the curve y=e^(2x)+x^2 drawn at the point x=0 is

A circle of radius 'r' passes through the origin O and cuts the axes at A and B,Locus of the centroid of triangle OAB is