Coordinate Geometry

Step 1: Given condition:
Step 2: Since is a polynomial, consider its degree. If , then in general, which leads to a contradiction.
Step 3: Hence, the only possible polynomial solution is a constant polynomial. Let
Step 4: Then which satisfies the given condition.
Step 5: Evaluate the given options using
Step 6: Hence, option (A) is correct.

Step 1: Let a point on the curve be where
Step 2: Distance of this point from the origin is: To minimize , minimize :
Step 3: Simplify:
Step 4: Differentiate with respect to : Set derivative equal to zero:
Step 5: Critical points:
Step 6: Evaluate at these points: For : For :
Step 7: Minimum distance:

Step 1: Combine into a single trigonometric function.
Step 2: Substitute in the integral:
Step 3: Use the standard integral:
Step 4: Here, .
Step 5: Therefore,

Step 1: Write the general term of the binomial expansion:
Step 2: Simplify the general term:
Step 3: Find the term containing . Coefficient of :
Step 4: Find the term independent of . Constant term:
Step 5: Find the required ratio: Using :
Step 6: Writing the ratio in the given form:

Step 1: Describe the set . This represents the closed disc of radius centered at the origin.
Step 2: Describe the set . This represents the region outside or on the circle of radius centered at the origin.
Step 3: Compare the two regions. There is no real number which is simultaneously and .
Step 4: Hence, no point can satisfy both conditions at the same time.

Step 1: Compare with standard circle
Step 2: Centre from coefficients:
Step 3: Radius squared:
Step 4: Condition . Solve inequality gives feasible integers count 16. Hence → (C).

Step 1: Find the equation and slope of line . Line cuts the axes at: Slope of :
Step 2: Write the equation of variable line . Since , its slope is: Let and . Slope of : Equating slopes:
Step 3: Find equations of lines and . Equation of through and : Equation of through and :
Step 4: Find coordinates of intersection point . Substitute into equation of : From : Equating both expressions for and simplifying, we eliminate to obtain:
Hence, the locus of point is

Step 1: Write the equation of the variable line through the origin. Let the line through origin be:
Step 2: Find point on the line . Substitute : Since the line passes through origin, distance is proportional to:
Step 3: Find point on the line . Substitute : Distance is proportional to:
Step 4: Find the ratio . Dividing by :
Hence, the point divides the segment in the ratio

Step 1: Substitute in curve:
Step 2: This becomes quadratic in x:
Step 3: For tangency, discriminant = 0:
Step 4: Solve gives two real values of c. Hence number = 2 → (C).

Step 1: Write the general second-degree equation of a conic. From the given equation: we identify: Step 2: Use the discriminant to identify the conic. The discriminant is: Step 3: Interpret the result.

If Parabola
If Hyperbola
If Ellipse (or circle if and )
Here, and , so the conic is an ellipse. Step 4: Final conclusion. The given equation represents an .

Step 1: Identify the order of the differential equation. The highest order derivative present in the equation is: Hence, Step 2: Remove radicals to determine the degree. Given equation: To remove the fractional power, square both sides: Step 3: Determine the degree. After removing radicals:

The equation is polynomial in derivatives
The highest order derivative is
Its highest power is
Therefore, Step 4: Final conclusion. The order and degree of the given differential equation are:

Step 1: Recall the basic requirement to form a triangle. To form a triangle in coordinate geometry, three non-collinear points are required. Step 2: Analyze the given data. Only two points are provided: Step 3: Check feasibility. With only two points:

A line segment can be formed
A triangle cannot be formed
No information about a third vertex is given. Step 4: Final conclusion. Since a triangle cannot be formed with only two vertices, none of the given properties apply.

Step 1: Rewrite the given equation in standard form. Bring all terms to one side: Divide throughout by : Step 2: Group and terms and complete the squares. Complete the square: Step 3: Compare with standard form. Thus, Step 4: Final conclusion. Centre Radius