Maths > Circles > 7.0 Length of intercept cutoff from a line by a circle or length of chord of a circle
Circles
1.0 Definition
2.0 Equation of circle in various forms
2.1 Central Form
2.2 Standard Form
2.3 Parametric Form
2.4 General form of circle
2.4.1 Application of general form of circle
2.5 Diametric form of circle
3.0 Intercepts made by a circle on coordinate axis
4.0 Position of a point with respect to a circle
5.0 Maximum and minimum distance of a point from a circle
6.0 Intersection of a line and a circle
7.0 Length of intercept cutoff from a line by a circle or length of chord of a circle
8.0 Equation of tangent to a circle
8.1 Slope form
8.2 Point form
8.3 If a point outside the circle is given through which tangent to a circle passes, then
8.4 Parametric form
9.0 Tangents from a point to the circle
10.0 Length of tangent from a point to a circle
11.0 Common Tangents
12.0 Equation of common tangents
13.0 Pair of tangents
14.0 Normal to a circle at a given point
15.0 Common chord of two circles
16.0 Equation of chord joining two points on circle
17.0 Equation of chord of circle whose midpoint is given
18.0 Chord of contact
19.0 Orthogonal Circles
20.0 Director Circle
21.0 Family of circles
7.1 Alternate Method
2.2 Standard Form
2.3 Parametric Form
2.4 General form of circle
2.4.1 Application of general form of circle
2.5 Diametric form of circle
8.2 Point form
8.3 If a point outside the circle is given through which tangent to a circle passes, then
8.4 Parametric form
Draw $OM$ perpendicular to $PQ$
$OM=$length of perpendicular from origin $(0,0)$ to line $y=mx+c$
$OM = \frac{{\left| c \right|}}{{\sqrt {1 + {m^2}} }}$[from previous topic of Intersection of a line and a circle]
$OP=$radius of circle$=a$
From figure $21$, In $\Delta OPM,PM = \sqrt {O{P^2} - O{M^2}} $
$$PM = \sqrt {{a^2} - \frac{{{c^2}}}{{1 + {m^2}}}} = \sqrt {\frac{{{a^2}\left( {1 + {m^2}} \right) - {c^2}}}{{1 + {m^2}}}} $$ $$PQ = 2PM = 2\sqrt {\frac{{{a^2}\left( {1 + {m^2}} \right) - {c^2}}}{{1 + {m^2}}}} $$
Question 13. The equation of circle is ${x^2} + {y^2} - 2x - 4y - 5 = 0$ and $y=x+3$ is the equation of chord to the circle. Find:
$(a)$ Length of chord
$(b)$ Middle point of chord
$(c)$ Angle subtended by a chord on major segment of the circle.
Solution: From the equation of circle, we get centre of circle $(1,2)$ and radius $r = \sqrt {{g^2} + {f^2} - c} = \sqrt { - {1^2} + - {2^2} + 5} = \sqrt {10} $
length of perpendicular from centre of circle $O(1,2)$ to line $y=x+3$ is $OP = \frac{{\left| {1 - 2 + 3} \right|}}{{\sqrt {{1^2} + - {1^2}} }} = \frac{2}{{\sqrt 2 }} = \sqrt 2 $
As shown in figure $22$, $OP = \sqrt 2 $, $OA = \sqrt {10} $
In $\Delta OPA$, apply Pythagoras theorem
$$O{A^2} = O{P^2} + P{A^2}$$ $${\sqrt {10} ^2} = {\sqrt 2 ^2} + P{A^2}$$ $$PA = 2\sqrt 2 $$ and $$PA = PB = 2\sqrt 2 $$
So, length of chord is $AB = 2PA = 4\sqrt 2 ...(a)$
The slope ${m_1}$ of chord $AB$ is $1$ and $OP$ is perpendicular to $AB$, so slope ${m_2}$ of line $OP$ can be find out using ${m_1} \times {m_2} = - 1$.
Therefore, ${m_2}=-1$
Equation of line $OP$ is $y-2=-1(x-1)$ or $x+y=3$
To find the intersection point $P$ of two lines $y=x+3$ and $y=-x+3$, solving them we get,
Coordinate of point $P$ is $(0,3)...(b)$
From figure $23$, $\tan \theta = \frac{{2\surd 2}}{{\surd 2}} = 2...(c)$
Question 14. The chord $y=x+3$ subtends an angle of ${30^ \circ }$ at major segment of circle ${x^2} + {y^2} = {k^2}$. Find the value of $k$.
Solution: As we know that the chord subtends an angle of ${30^ \circ }$ at major segment of circle, so the $\angle AOP = {30^ \circ }$ as shown in figure $24$.
Radius of circle $r = \sqrt {{k^2}} = k$
Length of perpendicular from centre of circle $O(0,0)$ to line $y=x+3$ is $$OP = \frac{{\left| {0 - 0 + 3} \right|}}{{\sqrt {{1^2} + - {1^2}} }}$$ $$ = \frac{3}{{\sqrt 2 }}$$ and $$\cos 30^\circ = \frac{{\frac{3}{{\sqrt 2 }}}}{k}$$
or, $$k = \sqrt 6 $$
