the point on the curve is 4y=x^2 nearest to (7,2) is:

this is what i did: i solved for y and I know that the there's a tangent line at (x1,y1).

so using y-y1=m(x-x1), I found the lines of the tangent and perpendicual lines:

slope is derivative so it is 1/2x1

tangent: y-y1=.5x1(x-x1)
perpendicualar: y-y1=-2/x1(x-x1)

then using system of equations i got (1/56, 784). i know that's wrong but i can't seem to know what i did wrong.

I also heard that you can do this problem by using the distance formula. can you explain that please

You have 4y=x^2 or y =(1/4)x^2, so the distance from the point to the curve is
D^2 = (x-7)^2 + (y-2)^2
substituting for y you get
D^2 = (x-7)^2 + ((1/4)x^2-2)^2 or
D= sqrt((x-7)^2 + ((1/4)x^2-2)^2)
Now find
dD/dx = 1/(2sqrt((x-7)^2 + ((1/4)x^2-2)^2)) * [2(x-7) + 2((1/4)x^2-2)*(1/2)x]
Setting this to 0 you get
[2(x-7) + 2((1/4)x^2-2)*(1/2)x]=0
which simplifies to
2x-14 + (1/4)x^3 -2x = 0 or
(1/4)x^3 = 14 and
x^3 = 56 so x = 56^(1/3)
See if this agrees with the method you're using.

14x(3)

It seems like there might be a misunderstanding in your calculations. Let's go through the steps again to find the correct answer.

You started by finding the equation of the tangent line using the slope of the derivative. The slope of the tangent line can be found by taking the derivative of the curve equation with respect to x. In this case, the curve equation is 4y = x^2, so the derivative is d/dx(4y) = d/dx(x^2). The derivative of 4y with respect to x is 4(dy/dx), and the derivative of x^2 with respect to x is 2x. Therefore, the slope of the tangent line is dy/dx = 2x/4 = x/2.

To find the equation of the tangent line, you correctly used the point-slope form of a line. However, there seems to be a mistake in the equation you obtained. The correct equation of the tangent line at the point (x1, y1) should be y - y1 = (x - x1)*(x/2).

Next, you tried to find the equation of the perpendicular line using the negative reciprocal of the slope of the tangent line. However, there's a mistake in the sign of the slope. The negative reciprocal of x/2 should be -2/x. Therefore, the correct equation of the perpendicular line is y - y1 = -2/x1*(x - x1).

You then solved the system of equations formed by the tangent and perpendicular lines to find the intersection point (1/56, 784). However, this appears to be incorrect. There might have been an error in solving the system of equations.

Now, let's explore how to solve the problem using the distance formula.

The point (x, y) on the curve is closest to the point (7, 2). The distance between these two points can be found using the distance formula:

d = sqrt((x - 7)^2 + (y - 2)^2).

To minimize the distance, we need to find the minimum value of d. Instead of differentiating and solving equations, we can minimize the distance directly by minimizing the square of the distance, which is equivalent.

D^2 = (x - 7)^2 + (y - 2)^2.

Substituting y = (1/4)x^2, we get:

D^2 = (x - 7)^2 + ((1/4)x^2 - 2)^2.

We want to find the value of x that minimizes D^2. To do this, we can find the critical points by setting the derivative dD^2/dx equal to zero.

dD^2/dx = 0.

Differentiating D^2 with respect to x, we have:

d/dx((x - 7)^2 + ((1/4)x^2 - 2)^2) = 0.

Expanding and simplifying, we get:

2(x - 7) + 2((1/4)x^2 - 2)(1/2)x = 0.

Simplifying further, we have:

2x - 14 + (1/4)x^3 - 2x = 0.

Combining like terms and rearranging, we obtain:

(1/4)x^3 = 14.

To solve for x, we can multiply both sides by 4:

x^3 = 56.

Taking the cube root of both sides, we find:

x = 56^(1/3).

By substituting this value of x back into the equation for y = (1/4)x^2, we can find the corresponding value of y.

Hence, the point on the curve nearest to (7, 2) is approximately (56^(1/3), y), where y = (1/4)(56^(1/3))^2.

I hope this clears up any confusion and helps you find the correct answer. Let me know if you have any further questions!