Home
Class 12
MATHS
A Line through the variable point A(1+k;...

A Line through the variable point `A(1+k;2k)` meets the lines `7x+y-16=0; 5x-y-8=0 and `x-5y+8=0` at B;C;D respectively. Prove that AC;AB and AD are in HP.

Text Solution

AI Generated Solution

To prove that the segments AC, AB, and AD are in Harmonic Progression (HP), we start by defining the points and the equations of the lines involved. ### Step 1: Define the Points and Lines Let the variable point \( A \) be given as \( A(1+k, 2k) \). The lines are defined as follows: - Line 1: \( L_1: 7x + y - 16 = 0 \) - Line 2: \( L_2: 5x - y - 8 = 0 \) - Line 3: \( L_3: x - 5y + 8 = 0 \) ...
Promotional Banner

Topper's Solved these Questions

  • STRAIGHT LINES

    CENGAGE ENGLISH|Exercise EXAMPLE|12 Videos

  • STRAIGHT LINES

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 2.1|23 Videos

  • STRAIGHT LINE

    CENGAGE ENGLISH|Exercise Multiple Correct Answers Type|8 Videos

  • THEORY OF EQUATIONS

    CENGAGE ENGLISH|Exercise JEE ADVANCED (Numerical Value Type )|1 Videos

Similar Questions

Explore conceptually related problems

A line through the variable point A(k+1,2k) meets the lines 7x+y-16=0,5x-y-8=0,x-5y+8=0 at B ,C ,D , respectively. Prove that A C ,A B ,A D are in HP.

Show that if any line through the variable point A(k+1,2k) meets the lines 7x+y-16=0,5x-y-8=0,x-5y+8=0 at B ,C ,D , respectively, the A C ,A B ,a n dA D are in harmonic progression. (The three lines lie on the same side of point Adot)dot

A straight line through the point A (-2,-3) cuts the line x+3y=9 and x+y+1=0 at B and C respectively. If AB.AC =20 then equation of the possible line is

A straight line through A (-15 -10) meets the lines x-y-1=0 , x+2y=5 and x+3y=7 respectively at A, B and C. If 12/(AB)+40/(AC)=52/(AD) prove that the line passes through the origin.

Prove that the three lines 2x-y-5=0, 3x-y-6=0 and 4x-y-7=0 meet in a point.

A straight line through origin O meets the lines 3y=10-4x and 8x+6y+5=0 at point A and B respectively. Then , O divides the Segment AB in the ratio.

The line y=(3x)/4 meets the lines x-y+1=0 and 2x-y=5 at A and B respectively. Coordinates of P on y=(3x)/4 such that PA*PB=25.

The orthocentre of the triangle formed by the lines x - 7y + 6 = 0, 2x - 5y - 6 = 0 and 7x + y - 8 = 0 is

Prove that the three lines 5x+3y-7=0, 3x-4y=10, and x+2y=0 meet in a point.

Find the point of intersection of the lines 2x-3y+8=0 and 4x+5y=6