Coordinate Geometry

If the curves and intersect perpendicularly, then we need to find the value of .

First, find the derivatives of each curve.

For , differentiate with respect to x:

. Let .

For , differentiate with respect to x:

. Let .

For the curves to intersect perpendicularly, the product of their slopes must be -1:

Now, we substitute into the equation :

Next, we substitute and into the equation :

Finally, we find :

Therefore, the correct option is (D) 8.

The two lines and are perpendicular if the product of their slopes is -1.

First, let's find the slopes of the lines.

For the line , we can rewrite it as , so . The slope, , is .

For the line , we can rewrite it as , so . The slope, , is .

For the lines to be perpendicular, .

So,

Therefore, the correct option is (A) .

The general equation of the plane meeting the coordinate axes at , , and is: Now, the centroid of the triangle formed by points , , and is given by: We are told that the centroid is at the point . So: From this, we get: Substituting these values into the general equation of the plane:

So, the correct answer is (B) : .

The centroid of a triangle is the point where the three medians intersect. It is also the average of the coordinates of the three vertices of the triangle. Since the given points are midpoints of the sides of the triangle, we can use these points to find the centroid. Let the three midpoints of the triangle be: The centroid is the average of the coordinates of the points: Therefore, the centroid of the triangle is .

Given:
- The equation of the straight line is passing through the point and is perpendicular to the line .
To find the equation of the line that satisfies these conditions, we use the fact that the general form of the equation of a line perpendicular to the given line can be derived from the condition of perpendicularity.
The given line has the equation:

The slope of this line is the negative reciprocal of the slope of the perpendicular line. To derive the equation of the perpendicular line, we use the point of intersection and the general formula for a line in the form .
After solving, the correct equation of the perpendicular line passing through the given point is:

Therefore, the correct answer is (B):

The correct answer is: (B): (2, 2).

We are given that the curve passes through the point and that at any point on the curve, the product of the slope of its tangent and the x-coordinate of the point is equal to the y-coordinate of the point. This can be written as:

Step 1: Set up the differential equation

The given relationship between the slope of the tangent and the coordinates of the curve is:

Step 2: Solve the differential equation

We now solve this first-order differential equation. We can separate the variables:

Now integrate both sides:

The integrals are straightforward:

Step 3: Solve for

Exponentiating both sides gives:

Thus, the general solution is:

Step 4: Use the given point to find

We are told that the curve passes through the point . Substituting and into the equation gives:

Thus, , and the equation of the curve is:

Step 5: Check for another point on the curve

Now, substitute into the equation to get:

Conclusion:
The curve passes through the point , so the correct answer is (B): (2, 2).