In the figure, $P A $ and $ P B $ are tangents from an external point $ P $ to a circle with centre $ O $. $ L N $ touches the circle at $ M $. Prove that $ P L+L M=P N+M N $. "
Given:
In the figure, \(P A \) and \( P B \) are tangents from an external point \( P \) to a circle with centre \( O \). \( L N \) touches the circle at \( M \).
To do:
We have to prove that \( P L+L M=P N+M N \).
Solution:
$PA$ and $PB$ are tangents to the circle from $P$.
This implies,
$PA = PB$
Similarly,
$LA$ and $LM$ are tangents from $L$.
$LA = LM$
$NB = NM$
Therefore,
$PA = PB$
$\Rightarrow PL + LA = PN + NB$
$PL + LM = PN + NM$
Hence proved.
Related Articles If \( a=x^{m+n} y^{l}, b=x^{n+l} y^{m} \) and \( c=x^{l+m} y^{n} \), prove that \( a^{m-n} b^{n-1} c^{l-m}=1 . \)
In the figure, if $l \parallel m, n \parallel p$ and $\angle 1 = 85^o$, find $\angle 2$."\n
From a point \( P \), two tangents \( P A \) and \( P B \) are drawn to a circle with centre \( O \). If \( O P= \) diameter of the circle, show that \( \Delta A P B \) is equilateral.
Simplify the following.a) \( (l^{2}-m^{2})(2 l+m)-m^{3} \)b) \( (p+q+r)(p-q+r)+p q-q r \)
If $l, m, n$ are three lines such that $l \parallel m$ and $n \perp l$, prove that $n \perp m$.
From an external point \( P \), tangents \( P A=P B \) are drawn to a circle with centre \( O \). If \( \angle P A B=50^{\circ} \), then find \( \angle A O B \).
In the figure, \( P A \) and \( P B \) are tangents to the circle from an external point \( P \). \( C D \) is another tangent touching the circle at \( Q \). If \( PA=12\ cm, QC=QD=3\ cm, \) then find \( P C+P D \)."\n
In the figure \( P O \perp Q 0 \). The tangents to the circle at \( P \) and \( Q \) intersect at a point \( T \). Prove that \( P Q \) and \( O T \) are right bisectors of each other."\n
If \( x=a^{m+n}, y=a^{n+1} \) and \( z=a^{l+m} \), prove that \( x^{m} y^{n} z^{l}=x^{n} y^{l} z^{m} \)
From an external point \( P \), tangents \( P A \) and \( P B \) are drawn to a circle with centre \( O \). At one point \( E \) on the circle tangent is drawn, which intersects \( P A \) and \( P B \) at \( C \) and \( D \) respectively. If \( P A=14 \mathrm{~cm} \), find the perimeter of \( \triangle P C D \).
In the figure, tangents \( P Q \) and \( P R \) are drawn from an external point \( P \) to a circle with centre $O$, such that \( \angle R P Q=30^{\circ} . \) A chord \( R S \) is drawn parallel to the tangent \( P Q \). Find \( \angle R Q S \)."\n
In the figure, \( P Q \) is a tangent from an external point \( P \) to a circle with centre \( O \) and \( O P \) cuts the circle at \( T \) and \( Q O R \) is a diameter. If \( \angle P O R=130^{\circ} \) and \( S \) is a point on the circle, find \( \angle 1+\angle 2 \)."\n
In the figure, there are two concentric circles with centre O. \( P R T \) and \( P Q S \) are tangents to the inner circle from a point \( P \) lying on the outer circle. If \( P R=5 \mathrm{~cm} \), find the length of \( P S \)."\n
In the figure, $p$ is a transversal to lines $m$ and $n, \angle 2 = 120^o$ and $\angle 5 = 60^o$. Prove that $m \parallel n$."\n
In the figure, $l, m$ and $n$ are parallel lines intersected by transversal $p$ at $X, Y$ and $Z$ respectively. Find $\angle l, \angle 2$ and $\angle 3$."\n
Kickstart Your Career
Get certified by completing the course
Get Started