Coordinate Geometry

Step 1: Using standard trigonometric identities.

Step 2: Simplifying using double angle formulas.

Step 3: Final simplification.

Step 4: Conclusion.

Step 1: Analyze the given condition.
We are given that: This implies a symmetry in the triangle. In any triangle, this condition can only hold if the angles are all equal and the sides are proportional. Step 2: Conclusion.
Since the angles are equal, the triangle is an equilateral triangle. Thus, the correct answer is .

Step 1: Condition for a pair of coincident lines.
For the equation to represent a pair of coincident lines, the condition is .
Step 2: Conclusion.
Thus, the condition for the equation to represent a pair of coincident lines is , corresponding to option (B).

Step 1: Rewrite the equation of the circle.
The equation of the circle is . Completing the square for and , we get: Thus, the radius of the circle is 5, and we find .
Step 2: Conclusion.
The correct answer is (D) .

Step 1: Compare with the standard form.
For a pair of straight lines through the origin: Here .
Step 2: Use the angle formula.
The angle between the lines is given by
Step 3: Substitute values.

Step 4: Identify .
Since , we get

Step 1: Understand the region.
The given circle is The region is bounded between and in the first quadrant.
Step 2: Write the area integral.
Area bounded by the curve and the x-axis is The factor 2 accounts for symmetry about the x-axis.
Step 3: Evaluate the integral.
Using the standard result with , we get Step 4: Substitute limits.

Step 1: Using the definition of probability from CDF.
For a discrete random variable,
Step 2: Applying the formula.

Step 3: Conclusion.
The required probability is .

Step 1: Rearranging the given equation.

Step 2: Grouping terms.

Step 3: Writing as a product of linear factors.

Step 4: Conclusion.
The separate equations of the lines are

Step 1: Use the parametric equations.
We are given the parametric equations and . To eliminate the parameter , we use the Pythagorean identity .
Step 2: Square and add the equations.
Squaring both and , we get: Adding these two equations, we get:
Step 3: Conclusion.
Thus, the Cartesian equation of the curve is , which corresponds to option (A).

Step 1: Recognize the equation of a pair of lines.
The general form of the equation of a pair of lines is: For the given equation , we can identify the coefficients:
Step 2: Use the formula for .
For a pair of lines, the formula for is given by: Substituting the values, we get:
Step 3: Conclusion.
Thus, the value of is 10, which corresponds to option (A).

Step 1: Equation of the line through two points.
The equation of the line passing through two points and is given by: Substituting and into the equation, we get: Simplifying, we obtain the equation of the line:
Step 2: Find the point of intersection.
Now, to find the point of intersection with the line , we substitute into this equation: Simplifying the equation: Substituting into , we get:
Step 3: Conclusion.
Thus, the point of intersection is , which corresponds to option (C).

Step 1: Write the two given equations.

Step 2: Eliminate the parameter .
From equation (1):
Step 3: Substitute in equation (2).

Step 4: Simplify the equation.
Multiplying throughout by :
Step 5: Identify the locus.
The obtained equation contains the product term , which represents a second–degree curve of hyperbolic nature.
Hence, the locus is a hyperbola.

Step 1: Using the formula for distance from a point to a line.
The distance of a point from the line is
Step 2: Substituting the given values.
Here, , , , and the point is .
Step 3: Solving for .

Step 4: Conclusion.
The values of are and .

Step 1: Use the condition for collinearity.
For three points to be collinear, the area of the triangle formed by them should be zero. The area can be calculated using the determinant method: Substitute the coordinates of , , and into this equation.
Step 2: Solve for .
After simplifying, we find that .
Step 3: Conclusion.
The correct answer is (A) 7.

Step 1: Recognize the parametric equations.
The given equations are in the parametric form of a circle. These represent a circle, with being the parameter.
Step 2: Find the radius and center of the circle.
For the parametric equations of the form the center is and the radius is given by Since can vary, we get the radius as .
Step 3: Conclusion.
Thus, the center is and the radius is units.

Step 1: Use of Polar to Cartesian conversion formula.
The formula for converting polar coordinates to Cartesian coordinates is: Given and , we compute:
Step 2: Conclusion.
Thus, the Cartesian coordinates are , which corresponds to option (D).

Step 1: Use the condition for a pair of lines through the origin.
Given equation represents a pair of straight lines through the origin. If is one line, then substituting it must satisfy the equation.

Step 2: Substitute .


Step 3: Simplify.


Step 4: Conclusion.

Step 1: Write the line in general form.
Given line: Multiplying by , Step 2: Use the perpendicular distance formula.
Distance from point to line is Step 3: Substitute values.
Step 4: Conclusion.
The required perpendicular length is

Step 1: Find the direction cosines of the lines.
We are given the direction ratios of two lines. For the first line, the direction ratios are , and for the second line, the direction ratios are .
Step 2: Use the formula for the angle between two lines.
The formula for the angle between two lines with direction cosines and is: Substitute the values and simplify to find .
Step 3: Conclusion.
Thus, the correct answer is , corresponding to option (D).

Step 1: Condition for a pair of lines.
For the equation to represent a pair of straight lines, the condition is , where , , and are constants in the equation.
Step 2: Conclusion.
Thus, the correct answer is , corresponding to option (B).

Step 1: General equation of the circle.
The general equation of a circle is: Since the circle passes through the origin, we substitute and into the equation: Thus, the equation becomes: Step 2: Apply the x- and y-intercepts.
We are given the x-intercept is 3 and the y-intercept is -5. For the x-intercept, set , and for the y-intercept, set . Substituting these values: For the x-intercept, : For the y-intercept, : Step 3: Final equation.
Substitute and into the equation: Step 4: Conclusion.
Thus, the equation of the circle is .

Step 1: Understand the requirement.
Lines parallel to the coordinate axes are of the form Their joint equation is

Step 2: Find the point of intersection of the given lines.
The given second-degree equation represents a pair of straight lines intersecting at a point . By comparing with the standard form and evaluating the intersection point, we obtain

Step 3: Form the joint equation.


Step 4: Conclusion.
Hence, the required joint equation is

Step 1: Use the relationships between the roots and coefficients.
Let the roots of the quadratic equation be and . The A.M. (Arithmetic Mean) and G.M. (Geometric Mean) are given by:
Step 2: Set up the equation using these relationships.
From the A.M., we have , and from the G.M., we have . The equation of the quadratic is: Substitute and :
Step 3: Conclusion.
The equation is .

Step 1: Write the parametric equations.


Step 2: Find the slope of the tangent.


Step 3: Condition for perpendicular to X-axis.
A tangent perpendicular to the X-axis is vertical, so

Step 4: Find the point of contact.
At :

Step 5: Conclusion.
The point of contact is

Step 1: Use the property of a circle.
If a point lies on a circle whose diameter endpoints are and , then which implies
Step 2: Find vectors and .

Step 3: Take dot product.

Step 4: Solve for .

Step 1: Completing the square.
To convert the given equation of the circle to a standard form, we complete the square for both and . The equation is: Completing the square for and , we get:
Step 2: Standard form of the circle.
The equation is now in the standard form of a circle: which represents a circle with center and radius 4.
Step 3: Parametric equations.
The parametric equations for a circle with center and radius are: Substituting , , and , we get:
Step 4: Conclusion.
Thus, the correct parametric equations are given by option (D).

Step 1: Finding the equation of the tangent.
For the curve , the derivative is: Thus: At the point , the slope of the tangent is:
Step 2: Equation of the tangent.
Using the point-slope form of the equation of a line, we have: Simplifying, we get:
Step 3: Conclusion.
Thus, the equation of the tangent is , which makes option (B) the correct answer.

Step 1: Find the midpoints of the diameter.
The center of a circle is the midpoint of the ends of its diameter. The midpoint of the given points of the diameters gives us the center of the required circle. Using the midpoint formula for the centers of the given circles, we obtain the center coordinates.
Step 2: Use the center and radius to write the equation of the required circle.
We now use the general form of the circle's equation and plug in the calculated center and radius. We arrive at the equation:
Step 3: Conclusion.
Thus, the equation of the required circle is , corresponding to option (D).

Step 1: Check continuity at .
To ensure the function is continuous at , the limit of as approaches 0 must equal . Let's calculate the limit:
Step 2: Calculate the limit.
By substituting , we get: Thus, .
Step 3: Conclusion.
The value of is .

Step 1: Solve the differential equation.
The given equation is . Taking the exponential of both sides, we get: This simplifies to:
Step 2: Separate variables and integrate.
Rewriting the equation: Integrating both sides gives: Using the initial condition when , we find .
Step 3: Conclusion.
Substituting the value of gives the solution:

Concept:
Two lines are perpendicular if their direction vectors are perpendicular: Step 1: Find direction ratios. From first line: Rewrite: Direction ratios: From second line: Rewrite: Direction ratios:
Step 2: Apply perpendicular condition.
Step 3: Simplify. Multiply by 15:
Step 4: Find required expression. Multiply both sides appropriately:
Step 5: Conclusion.

Step 1: Convert to standard form. Circle 1: Centre , radius Circle 2: Centre , radius
Step 2: Distance between centres.
Step 3: Compare. So circles intersect.
Step 4: Conclusion. Number of common tangents = 3

Step 1: Find Normal Vector
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Step 2: Equation of Plane
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Plane: .
Step 3: Distance from Origin
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Wait, let's re-calculate : . Correct. Result is (A).
Final Answer: (A)

We are given a geometric problem that involves both trigonometric functions and coordinate geometry.
Step 1: Analyze the given equations We have the equation: Simplifying the equation: Multiplying through by 6 to clear the denominators: Simplifying further: This gives a linear equation involving and , likely corresponding to a line equation in the coordinate plane.
Step 2: Relating the trigonometric functions to the area We are also given the trigonometric equation: This equation could relate to the angles of the triangle. Typically, this type of relation can help find the area of the triangle through trigonometric identities. However, we are already told that the area of is , and thus we do not need to solve for it explicitly.
Step 3: Conclusion Thus, the area of the triangle is .

We are tasked with finding the area of triangle , given the equation involving trigonometric functions and the condition that the coordinates satisfy a linear equation.
Step 1: Simplify the given linear equation We are given the equation: Simplify the equation: Multiply through by 6 to eliminate the denominators: Simplify further: This gives a line equation, which could represent one of the sides of triangle .
Step 2: Analyze the trigonometric part We are also given the equation involving trigonometric functions: This equation may be used to find the relationship between the angles and , but as per the problem setup, it seems the area of is directly provided as .
Step 3: Conclusion Since the problem directly provides that the area of triangle is 2, the solution to the problem is:

Concept:
Compare coefficients of vectors. Step 1: Add LHS.
Step 2: Compare RHS.
Step 3: Match coefficients. Thus:
Step 4: Conclusion. Values =

Step 1: Evaluate statements.

• : False (counterexample exists)
• : False (sum of squares )
• : True (increasing in given interval)

Step 2: Check options. (C):
Step 3: Conclusion. Correct option is (C)

Step 1: Understanding the Concept:
We use the property that for . This simplifies the series significantly.

Step 2: Key Formula or Approach:
1. .
2. (used here for subtraction).

Step 3: Detailed Explanation:
Let the LHS be :
Using :
The equation is .
Since :
Using the formula :
By comparison:

Step 4: Final Answer:
The value of is .

Step 1: Understanding the Concept:
We are given a composite function . We need to apply the Chain Rule: .

Step 2: Key Formula or Approach:
1. .
2. .
3. using the Quotient Rule.

Step 3: Detailed Explanation:
Let . Then .
Let . Then .
Calculate :
Now, .
Substituting back into the chain rule expression:
Comparing with :
.

Step 4: Final Answer:
The values are .

Step 1: Understanding the Concept:
The internal angle bisector of a triangle divides the opposite side in the ratio of the sides containing the angle.
In , the angle bisector of divides the side in the ratio .

Step 2: Key Formula or Approach:
1. Length of side .
2. Length of side .
3. Internal Section Formula: If divides in ratio , then .

Step 3: Detailed Explanation:
The ratio in which divides is .
Let and .
Wait, the ratio is .
Applying the section formula for internal division:
Since , we can write and .
Substituting the vector magnitudes:
Rearranging for standard order:

Step 4: Final Answer:
The position vector of is .