Question

A straight line passes through the points (a,0) and (0,b).The length of the line segment contained between the axes is 13 and the product of the intercept on the axes is 60.Calculate the values of a and b and find the equation of the straight line​

Answer 1

SOLUTION :

GIVEN

• A straight line passes through the points (a,0) and (0,b)

• The length of the line segment contained between the axes is 13

• The product of the intercept on the axes is 60

TO DETERMINE

• The values of a and b

• The equation of the straight line

EVALUATION

Here the straight line passes through the points (a,0) and (0,b)

So the equation of the line is

$\displaystyle \sf{} \frac{y - 0}{x -a } = \frac{b - 0}{0 - a}$

$\implies\displaystyle \sf{} \frac{y}{x -a } = \frac{b }{ - a}$

$\implies \displaystyle \sf{} bx - ab = - ay$

$\implies \displaystyle \sf{} bx + ay = ab$

$\implies \displaystyle \sf{} \frac{x}{a} + \frac{y}{b} = 1 \: \: \: \: .....(1)$

This is in intercept form

So the line intersect x axis at A ( a, 0) and y axis at B (0,b)

Now the length between A & B

$\sf{} = \sqrt{ {(a - 0)}^{2} + {(0 - b)}^{2} }$

$\sf{} = \sqrt{ {a}^{2} + { b}^{2} }$

By the given condition

$\sf{} \sqrt{ {a}^{2} + { b}^{2} } = 13$

$\implies \sf{} {a}^{2} + { b}^{2} = 169 \: \: \: ...(2)$

Also The product of the intercept on the axes is 60

So

$\sf{}ab = 60 \: \: \: ....(3)$

Squaring equation (3) we get

$\sf{} {a}^{2} {b}^{2} = 3600$

Using Equation (2) we get

$\sf{} {a}^{2} (169 - {a}^{2} ) = 3600$

$\implies \sf{} {a}^{4} - 169 {a}^{2} + 3600 = 0$

$\implies \sf{} {a}^{4} - 144 {a}^{2} + - 25 {a}^{2} + 3600 = 0$

$\implies \sf{} {a}^{2} ( {a}^{2} - 144 ) - 25( {a}^{2} - 144) = 0$

$\implies \sf{} ( {a}^{2} - 144 )( {a}^{2} -25) = 0$

$\sf{} So \: either \: ( {a}^{2} - 144 ) = 0 \: or \: ( {a}^{2} -25) = 0$

$\sf{}( {a}^{2} - 144 ) = 0 \: gives \: a = \pm \: 12$

$\sf{}( {a}^{2} - 25 ) = 0 \: gives \: a = \pm \: 5$

Using Equation (3)

$\sf{}When \: \: \: a = 12 \: , \: b = 5$

$\sf{}When \: \: \: a = - 12 \: , \: b = - 5$

$\sf{}When \: \: \: a = 5 \: , \: b = 12$

$\sf{}When \: \: \: a = - 5\: , \: b = - 12$

Which are the required values of a and b

Now required equation of the line is any one of the below :

$\displaystyle \sf{}1. \: \: \: \: \frac{x}{12} + \frac{y}{5} = 1$

$\displaystyle \sf{}2. \: \: \: \: \frac{x}{ - 12} + \frac{y}{ - 5} = 1$

$\displaystyle \sf{}3. \: \: \: \: \frac{x}{5} + \frac{y}{12} = 1$

$\displaystyle \sf{}4. \: \: \: \: \frac{x}{ - 5} + \frac{y}{ - 12} = 1$

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LEARN MORE FROM BRAINLY

Derive the equation of a line having X and Y intercept value as 'a' and 'b' respectively

and hence find the equation of the line

https://brainly.in/question/24598684