Question

in triangle abc the equation of altitudes am and bn are x+5y-3=0, x+y-k=0. if the altitude is given by 3x-y-1=0 then find k

Answer 1

Complete question:

In triangle ABC the equation of altitudes AM and BN are x+5y-3=0, x+y-k=0. if the altitude CL is given by 3x-y-1=0 then find k.

Answer:

The value of k is 1.

Step-by-step explanation:

Given,

The equation of AM is x + 5y - 3 = 0

The equation of BN is x + y - k = 0

The equation of CL is 3x - y - 1 = 0

To find,

The value of k in the equation x + y - k = 0

Calculation,

We first try to find out the intersection point of the equations AM, BN, and CL.

For that, we try to solve the equations AM and CL.

i.e. x + 5y = 3....(1), and 3x - y = 1...(2)

But from equation (1) we can write:

x = 3 - 5y...(3)

We substitute equation (3) in equation (2), and we get:

3(3 - 5y) - y = 1

⇒ 9 - 15y - y = 1

⇒ y = 1/2

and x = 3 - 5(1/2) = 1/2

i.e. x = 1/2.

Now as the equation BN also passes through the point of intersection (1/2, 1/2), so we substitute.

x + y - k = 0

⇒1/2 + 1/2 - k = 0

⇒ k = 1

Therefore, the value of k is 1.

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