Question

find the equation of the line joining the point (2,-9) and the point of intersection of lines 2x+5y-8=0 and 3x-4y-35=0​

Answer 1

In the above Question , the following information is given -

We have to find the equation of the line joining the point (2,-9) and the point of intersection of lines 2x + 5y - 8 = 0 and 3x - 4y - 35 = 0 .

Solution -

Here , we have the following Equations of the lines given as -

Line 1 => 2x + 5y - 8 = 0

Line 2 => 3x - 4y - 35 = 0

Now , to find the required point of Intersection , we have to solve these two equations .

Equation 1 -

=> 2x + 5y = 8

Equation 2 -

=> 3x - 4y = 35

Plotting these on a line graph , we get -

The required Intersection point is ( 9, -2 ) .

See the graph in the attachment for more details .

So , finally , we have got two points through which this line passes -

Point 1 => ( 2, -9 )

Point 2 => ( 9, -2 )

Slope of the above line -

=> [ y² - y¹ ] / [ x² - x¹ ]

=> [ - 2 + 9 ] / [ 9 - 2 ]

=> [ 7 / 7 ]

=> 1

According to the slope intercept form -

y - y¹ = m ( x - x¹ )

As , m , the slope is 1 , this reduces to -

y - y¹ = x - x¹

x¹ = 2

y² = -9

So ,

y - ( - 9 ) = x - 2

=> y + 9 = x - 2

=> x - y - 11 = 0 .

Thus , the required equation is x - y - 11 = 0 .

This is the required answer .

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Note -

Here , I have used the exponential notations to denote the various points .

x¹ stands for x_1 or 2

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Answer 2

$\large\underline{\underline{\pink{\sf Answer}}}$

☞ You Equation is x-y-11 = 0

$\large\underline{\underline{\red{\sf Solution}}}$

We are given,

➝ 2x+5y-8 = 0

➝ 3x-4y-35 = 0

On Solving these Equations

➳ 2(9)+5(-2)-8 = 0

➳ 18-10-8 = 0

➳ 0 = 0

➳ x = 9 and y = -2

Similarly,

➳ 3(9)-4(-2)-35 = 0

➳ 27+8-35 = 0

➳ 0 = 0

➳ x = 9 and y = -2

Therefore the Intersection point is (9, -2)

Now that we found two points through which they pass, that is, (2, -9) and (9, -2)

Slope of them is given by,

➢ (y² - y¹)/(x² - x¹)

➢ {(-2)+9}/{9-2}

➢ 1

As per the slope intercept form,

»» y - y¹ = m( x - x¹ )

The slope is 1 so this reduces to

»» y - y¹ = x - x¹

»» x¹ = 2

»» y² = -9

Finally,

≫ y-(-9) = x-2

≫ y+9 = x-2

≫ x-y-11 = 0 [Ans]

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