Home
Class 10
MATHS
Write with proof which of the chords...

Write with proof which of the chords passing through any point in a circle will be the least .

Text Solution

Verified by Experts

Infinite number of chords can be drawn through a point in a circle . Among them the chord of which the line segment obtained by joining the centre and the internal point is a perpendicular bisector will be the least in length . Because we know that so far as the pependicular distance of the chords from the centre increases, the lenght of the chord proportionately diminishes and among the chords which can be drawn through this point , the perpendicular distance form the centre of that chords will be the greatest, i.e., the perpendicular distance of that chord from the centre will be the greatest .
Hence the lenght of that very chord will be the leat .
Mathematically let p be any point in the cirlce with centre at O. Let us join O, P let us drawn a line segment CD such that CD is perpendicular to OP at P and which intersects the circle at C and D. Then CD is a chords passing thorough P . Let AB be another chord passing though P.
we have to prove that the chord CD is smaller .
Construction : Let us draw ` OQ bot AB`
Proof ` angle OQP =1` right angle `[because OQ bot AB]`
In `Delta OPQ gt angle OPQ rArr OP gt OQ`
[ `because` the side opposite to greater angle is greater that the side opposite to smaller angle ] . i.e., the perpendicular distance of the chord CD form the centre O is greater that of the chord AB.
So, the chord CD is at a greater distance than that of AB from the centre .
Promotional Banner

Topper's Solved these Questions

  • THEOREMS RELATED TO CIRCLE

    CALCUTTA BOOK HOUSE|Exercise Example (B. Write true or false )|3 Videos

  • THEOREMS RELATED TO CIRCLE

    CALCUTTA BOOK HOUSE|Exercise Example (C. Fill in the blanks|2 Videos

  • THEOREMS RELATED TO CIRCLE

    CALCUTTA BOOK HOUSE|Exercise Example (Short-Answer Type Questions)|5 Videos

  • SPHERE

    CALCUTTA BOOK HOUSE|Exercise EXERCISE|36 Videos

  • THEOREMS RELATED TO CYCLIC QUADRILATERAL

    CALCUTTA BOOK HOUSE|Exercise Long -answer type questions (L.A.)|12 Videos

Similar Questions

Explore conceptually related problems

How many chords can be drawn through 21 points on a circle?

In two circles , one circle passes through the centre O of the other circle and they intersect each other at the points A and B . A straight line passing through A intersect the circle passing through O at the point P and the circle with centre at O at the point R . BY joining P , B and R , B prove that PR= PB.

AB and AC are two tangents to the circle with centre at O. Prove that AO bisects the chord passing through point of contact at right angle.

P is any point inside the circle with centre at O. If the radius of the circle be 5 cm and OP = 4 cm, then determine the length of the chord passing throgh P and which is also the least in length.

Find the equation of the chord of the circle x^2+y^2=a^2 passing through the point (2, 3) farthest from the center.

The tangent to a circle and the radius passing through the point of contact are perpendicular to each other.

Statement - I : The all chords passing through focus of an ellipse , the latus rectum will be the minimum in length . Statement - II : The sum of the reciprocals of the segments of any focal chord of an ellipse Is half of latus rectum .

OA is the radius of a circle with centre at Q, AQ is its chrod and C is any point on the circle. A circle passes through the point O,A,C intesect the chrod AQ at the point P. Prove that CP=PQ

The radius of the circle passing through the centre of incircle of DeltaABC and through the end points of BC is given by

Find the equation to the locus of mid-points of chords drawn through the point (0, 4) on the circle x^(2) + y^(2) = 16 .