If line y = x3 + 2 intersects line y = x2 + 3 at point A, what are

y=x√3+√2 intersects line y=x√2+√3

Let it intersect at point P
Solving both the equations
Eq. 1 : y - x√3 - √2
Eq. 2: y - x√2 - √3

Subtracting Eq. 2 from 1

We get x =1
y = √2 + √3

IMO D

y = sqrt(3)x + sqrt(2)
y = sqrt(2)x + sqrt(3)
sqrt(3)x + sqrt(2) = sqrt(2)x + sqrt(3) => x = 1
Then y = sqrt(3) + sqrt(2)

Ans = D

At point of insertion of two lines, x and y coordinates are equal. Hence equating both lines we can find the value of x coordinate.

x√3+√2=x√2+√3
x√3-x√2=√3-√2
x(√3-√2)=√3-√2
x=1

Putting value of x in any equation of line will give us the value of y coordinate.
y=x√3+√2
y=(1)√3+√2
y=√3+√2

Hence the coordinated of point A are (1, √3+√2).

IMO (d) is the correct option.

The easiest way to solve the equations is to put the option.

Pick options A, and D first as 'y' and 'x' co-ordinate is easy to check for 'x' and 'y.

Option D satisfies the equations.

Answer D

Even without answer choices, just looking at \(y = x√3 + √2\) and \(y = x√2 + √3\), clearly y will be the same when x = 1, so (1, √3 + √2) is on both lines.

y=x√3+√2 and y=x√3+√2 intersects line y=x√2+√3 at point A -> implies -> x√3+√2 = x√3+√2
-> √3(x-1) = √2(x-1)
Now, if (x-1) can be divided with each other, that would make √3 = √2, which we know cannot be true. So, (x-1) cannot be divided -> implies -> x-1 = 0; x = 1

For y, substitute x in any of the line equations.
Ans -> D.

Hey guys!

When you have two lines intersecting, the point of intersection must satisfy the equation for both lines

At that point, both lines exist and both line equations must be valid at the same time

Always take a quick glance through all the answer choices to see if there's an immediate fit

If you don't see anything, start plugging in the answer choices and see which one works. Start with the simpler ones because, hey you never know


B.

\(y = x\sqrt{3} + \sqrt{2}\)

\(\sqrt{2} = \sqrt{3}(\sqrt{3}) + \sqrt{2} = 3 + \sqrt{2}\)

This is wrong


C. \(\sqrt{3} = \sqrt{2}\sqrt{3} + \sqrt{2}\\
\)

This is wrong


D. \(\sqrt{3} + \sqrt{2} = (1)\sqrt{3} + \sqrt{2}\)

This one works! Let's try it for the other line


\(\\
y = x\sqrt{2} + \sqrt{3}\)

\(\sqrt{3} + \sqrt{2} = (1)\sqrt{2} + \sqrt{3}\)

This one works for both lines! (D) is the correct answer

It can be spotted by seeing that 1 as the x is easy to multiply and check the equations against the y

Do we coordinate geometry in the Focused Edition??

Though hardcore geometry isn’t tested on the GMAT Focus, some parts of coordinate geometry and functions still are. However, I've never seen any coordinate geometry questions like that in GMAT Prep mocks or the new Official Guides, so it's unlikely you'll encounter them on the test.