Home
Class 12
MATHS
Prove that quadrilateral A B C D , where...

Prove that quadrilateral `A B C D` , where `A B-=x+y-10 ,B C-=x-7y+50=0,C D-=22 x-4y+125=0, a n d D A =2x-4y-5=0,` is concyclic. Also find the equation of the circumcircle of `A B C Ddot`

Text Solution

Verified by Experts

If ABCD is a cyclic quadrilateral, then the equation of circumcircle of ABCD will be
`(x+y-10)(22x-4y+125)+lambda(x-7y+50)(2x-4y-5)=0`
Since it represents a circle, we have
Coefficient of `x^(2)=` Coefficient of `y^(2)`
`implies 22+2 lambda= -4+28 lambda`
`implies lambda =1`
Also, coefficient of xy is zero
`:. 22-4+lambda(-4-14)=0`
`:. lambda=1`
Thus, value of `lambda` from two conditions is same.
Therefore, ABCD is cyclic quadrilateral.
Also, equation of circle is
`(x+y-10)(22x-4y+125)+(x-7y+50)(2x-4y-5)=0`
`:. 24x^(2)+24y^(2)-1500=0`
or `2x^(2)+2y^(2)=125`
Promotional Banner

Topper's Solved these Questions

  • CIRCLE

    CENGAGE ENGLISH|Exercise Examples|13 Videos

  • CIRCLE

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 4.1|1 Videos

  • BINOMIAL THEOREM

    CENGAGE ENGLISH|Exercise Matrix|4 Videos

  • CIRCLES

    CENGAGE ENGLISH|Exercise Comprehension Type|8 Videos

Similar Questions

Explore conceptually related problems

In triangle A B C , the equation of side B C is x-y=0. The circumcenter and orthocentre of triangle are (2, 3) and (5, 8), respectively. The equation of the circumcirle of the triangle is a) x^2+y^2-4x+6y-27=0 b) x^2+y^2-4x-6y-27=0 c) x^2+y^2+4x-6y-27=0 d) x^2+y^2+4x+6y-27=0

In a cyclic quadrilateral A B C D , /_A=(2x+4)o,\ \ /_B=(y+3)o,\ \ /_C=(2y+10)o,\ \ /_D=(4x-5)o . Find the four angles.

In a cyclic quadrilateral A B C D , /_A=(2x+4)^o,\ /_B=(y+3)^o,\ /_C=(2y+10)^o,\ \ /_D=(4x-5)^o Find the four angles.

Find the four angles of a cyclic quadrilateral A B C D in which /_A=(2x-1)^circ , /_B=(y+5)^circ\ \ /_C=(2y+15)^circ and /_D=(4x-7)^circ .

In a parallelogram A B C D , if /_A=(3x-20)^0,\ /_B\ (y+15)^0,\ /_C=(x+40)^0, then find the values of x\ a n d\ y

A B C D is a cyclic quadrilateral in which A C\ a n d\ B D are its diagonals. If /_D B C=55^0\ a n d\ /_B A C=45^0, find /_B C D

If the lines x+a y+a=0,\ b x+y+b=0\ a n d\ c x+c y+1=0 are concurrent, then write the value of 2a b c-a b-b c-c adot

The equations of the sides A B ,\ B C\ a n d\ C A\ of\ "triangle"A B C\ a r e\ y-x=2,\ x+2y=1\ a n d\ 3x+y+5=0 respectively. The equation of the altitude through B is x-3y+1=0 b. x-3y+4=0 c. 3x-y+2=0 d. none of these

In Fig.85, A B|| C D\ a n d\ A C || B Ddot Find the value of x , y , zdot

In a triangle A B C , if A is (2,-1),a n d7x-10 y+1=0 and 3x-2y+5=0 are the equations of an altitude and an angle bisector, respectively, drawn from B , then the equation of B C is (a) a+y+1=0 (b) 5x+y+17=0 (c) 4x+9y+30=0 (d) x-5y-7=0